Integral Equations 0378-620X/89/020155-0851.50+0.20/0 and Operator Theory (e) 1989 Birkh~user Verlag, Basel Vol. 12 (1989)

A GENERALIZATION OF A THEOREM OF AMEMIYA AND ANDO ON THE CONVERGENCE OF RANDOM PRODUCTS OF CONTRACTIONS IN HILBERT SPACE

John Dye

Let {T 1, ..., TN} be a finite set of linear contraction mappings of a Hilbert space Hinto itself, and let r~oe a mapping from the natural numbers N to { 1, ..., N} which assumes each value infinitely often. One can form S n = Tr(n) "'" Tr(1) which could be described as a random product of the Ti's. If the contractions have t'h6 condifibn (W): IlWxll < I~1 whenever Tx ~ x, then S n converges weakly to the projection Q onto the subspace ni_~l~ [ x I Tix ='x]. This theorem is due to Amemiya and Ando. We demonstrate a basic prbperty of the algebraic semigroup S = S(T 1 ..... TN) generated by N contractions, each having (W). We prove that if the semigroup of an infinite set of contractions is equipped with this property, and the maps satisfy a minor condition parallel to (W) on each of N maps, then random products still converge weakly. Our proof is different from Amemiya and Ando's. We illustrate our method with a new proof of the fact that if a contraction T is completely non-unitary, then T n ---) 0 weakly.

1. INTRODUCTION. If P and Q are orthogonal projections onto subspaces Sp and SQ, respectively, in a Hilbert space H, then the sequence of products P, QP, PQP, QPQP, ... converges strongly to the projection onto the subspace Sp n SQ. This was known to J. yon Neumann in 1933; see [6], More generally, letting {P1, P2 ..... PN} be a finite set of projections, and r a mapping from N to {1,2 ..... N}, we can form S n = Pr(n) "'" Pr(2)Pr(1) This is called a random product of the Pi's. Independently, F. Browder (giving credit to S. Kakutani) ([ 2], 1958) and M. Prager ([7], 1960) proved the following: suppose that the selection r is quasi-periodic, in the sense that, for some fixed k > 0, each block of k consecutive values of r contains all values 1, 2, ..., N. Then S n converges weakly to the orthogonal projection Q onto the subspace of vectors fixed under each Pi" In 1965, Amemiya and Ando [1] proved that the periodicity assumptions could be 156 Dye dropped, and moreover, that only the condition (W) : IlTxll < Ilxll whenever Tx # x, on each of N contractions, is sufficient for weak convergence of the random product. Evidently, condition (W) appears to have been first used by Halperin ([4], 1962). It is of interest to note that results in the Amemiya and Ando paper have been applied to computer tomography. (See [9].) Amemiya and Ando's proof involved induction (on the finite number of maps.) Although ingenious, it was complicated by the fact that the "new map" can occur anywhere in a given sequence, an infinite number of times! It is natural to ask if a different style of proof, necessarily not based on induction, might accommodate random products from an infinite set of contractions. Our approach reveals a striking uniform condition on the algebraic semigroup S generated by a finite number of (W) contractions, which we shall call (USC): for any sequence of bounded vectors Vn---) 0 weakly, and any sequences of letters (Tn) and words (Wn) from S, Tn*Ynvn~ 0 weakly, and, if [[Vnl[ - [IWn Vnll --~ O, then W n v n ~ 0 weakly, also. Trivial counterexamples (to weak convergence) exist when the maps in the random product come from an infinite set of projections. However, in the presence of (USC), we show that a modest variation of the assumption of (W) for each of N maps to an infinite set of maps gives weak convergence of the product. ff a contraction T satisfies ( [[Tnx[[ = IlT*nx[[ = Ilx[[for all n > 0 imples x = 0) it is called completely non-unitary. (See Sz.-Nagy and Foias [5].) We will designate this class by (CN). In the next section, we demonstrate our technique for the main theorem by showing that a product of a single (CN) map converges weakly to zero. This was probably first proved by Foguel [3]. A different proof appears in Amemiya and Ando [1].

2. THE WEAK CONVERGENCE OF A SINGLE (CN) CONTRACTION. LEMMA 1. Let (Xn) be a bounded sequence of veclors in a Hilbert space which does not converge weakly to zero. By dropping to an appropriaIe subsequence, we may assume that Xn = ff-n u + v n, where u ~ 0, t/n---> 1, u _k v n V n, and v n ---> 0 weakly. Proof. Since (Xn) does not converge weakly to zero, there exists some w ~ H, some a > 0, and for some subsequence we may assume (2.1) [(xn, w)l > a. Let SUPn ( IlXnll ) = b < oo. By the weak compactness of B(0; b) we can drop to a convergent further subsequence of (Xn) and assume (2.2) x n ~ u weakly (which by (2.1) is not zero). Dye 157

Since H = < u > @ < u > _1_, we can decompose x n = e.nu + Vn, where v n < u >2_. To prove the Lemma, it suffices to show that CZn---~ 1 and v n --~ 0 weakly. Let 0 ~ z H be arbitrary, and decompose z as above by writing z = c~u + 13 v, where v < u > _k. (2.2) implies (x n - u, z) = (~nu + v n - u, ocu + 13v) = (2.3) ((~a- 1 ) u + v n, czu + 13v ) ---> 0. If 13 = 0 then (2.4) (Vn, cz u) = 0 and (2.3) gives cz(ct,n - 1) Ilull2 ~ 0, proving that or.n ---) 1. Ifo~ = 0 then (2.3) gives

((%-l)u+v n,13v) = (2.5) (v n, 13v) ---> 0. Hence (Vn, z)= (v n,czu+13v) = (vla, czu) + (v n,13v) = 0 + (v n,13v) --) 0 by (2.4) and (2.5). Thus v n ~ 0 weakly. PROPOSITION 1. (Foguel [3]). Let T have condition (CN). Then T n converges weakly to zero. Proof. We assume for some x that T n x does not converge weakly to zero, and arrive at a contradiction. By Lemma 1 we can drop to a subsequence and assume Tnx = crn u + v n, where u # 0, r n ~ 1, v n ~ 0 weakly and u .1_ v n. Note that IlTnxl[, lITn-k xf[, and liTn+k xl[ must all share the same nonzero limit L = ( I[ull2 + lim n [[Vnl]2)1/2. Let k _> 0 be fixed. We have (2.6) (Tnx, Tnx) = (Tn-kx, T *k (o~n u + Vn) ) and (2.7) (Wn+k x, TkTnx) = (Tn+kx, T k (czn u + Vn)). Applying the Schwarz inequality to the right hand side of (2.6) and (2.7), and taking limits as n ~ ~, we get by v n ----r0 weakly (2.6') L < ( lIT*k ul[ 2 + lim SUPn [[T*kvn[[ 2 )1/2 and (2.7') L _ ( lit k ull 2 + lim SUPnl[Tk v n II2 )112 Since both expressions on the right hand sides of (2.6') and (2.7') are < L, we must have that itT*kutl = I/utl and ilTku[I = Ilufl. Since k was arbitrary and T has condition (CN) we conclude that u must be zero, contradicting our assumption that Tnx does not converge weakly to zero.

3. RANDOM PRODUCTS OF CONTRACTIONS. PROPOSITION 2. Let S = S(T 1 ..... TN) be the algebraic semigroup generated by N linear contractions in a Hilbert space, each having (W). Then S has (USC). Proof. We note first that since T n can take on at most N distinct maps, the condition Tn*T n Vn--> 0 is satisfied for (bounded) vectors v n ---> 0. Let Q be the projection onto the subspace NQ = ~ kl'I=l F(Tk). Then we can decompose the v n as Vn(1) + Vn(2), where 158 Dye

Vn(1) a QH and Vn(2) e (I - Q)H. Expanding (3.1) Ilvnll2- IlWnvnll2, decomposing v n, and recalling that a vector fixed by a contraction is also fixed by its adjoint (see [7]), we see that (3.1) ~ 0 implies that Ilvn(2)l[2 - I[Wnvn(2)[I 2 ~ 0. Hence if (USC) holds on (I - Q)H, then the condition IlvnlI - ]lWnvn[ I ~ 0 will imply Wnv n ~ 0 weakly on all of H, as (USC) holds automatically on QH. To complete the proof, we will replace (I - Q)H by H and assume NQ -- (0). We proceed by induction, considering the case N = 1 first. Suppose that Tk(n) v n does not converge weakly to zero, for some bounded v n --> 0 weakly and some map k(n): N to N, even though Ilvnll - IITK(n) vnll ~ 0. We can drop to a subsequence and assume k(n) > 1. Necessarily Tk(n) -1 v n does not converge weakly to zero, so we can employ Lemma 1, and for a subsequence assume (3.2) Tk(n)-I Vn = anu + qn where u ~ 0, oqa---> 1, u .1_ qn Vn, and qn---> 0 weakly. We now compute (3.3) IITk(n)-I v n II2 - Ilwk(n)vnll 2 = (3.4) I[O~nUlL2- IIU.nTull2 + (3.5) [Iqnll2- IlTqnl[2 + (3.6) (qn, ~ u) + (0tn u, qn) (Tqn, a n Tu) (o~n Tu, Tqn ). As qn --* 0 weakly, (3.6) ~ 0. Because [[vnll - IlTk(n) Vn[I ~ 0, we note that (3.3) ~ 0. Hence (3.4) ~ 0 (as (3.5) is nonnegative), which implies, because ctn ~ 1, that IITull = IlulL. As T has (W), we conclude that Tu = u, contradicting (3.2) u ~ 0, as we are assuming NQ = (0). Hence T k(n) Vn--->0 weakly. Proceeding inductively, suppose the Proposition is valid for words from k-1 < N letters. Say there exist bounded vectors v n ~ 0 weakly and words W n e S(T 1 ..... T k) such that Wnv n does not converge weakly to zero, despite (NQ = ~ k=l F(Tj) = (0)) and (3.7) IlVnll- //Wnvnl/ ---> 0. By induction and a possible drop to a subsequence, we may assume that the Wn's are complete (each contains all k letters), and that (3.8) Wnv n ~ z weakly, for some z # 0. An easy combinatorial argument (and a possible reumeration of the Tj's, j = 1..... k) allows us to assume that (for some subsequence) there exist words F n T k and B n such that W n = FnTkBn, where the F n E S(T 1 ..... Tk_l) and are complete (in T 1 ..... Tk.1). We note that B n v n does not converge weakly to zero. If it did, then TkB n v n would converge weakly to zero, and by induction, (since IlTkBnVnll - IlFnTkBnVn[[ ~ 0 by (3.7)), FnTkBn v n would also Dye 159

converge weakly to zero, contradicting (3.8). Applying Lemma 1 to B n v n, we may assume that (3.9) BnVn = tY'nU + qn, where u ~ 0, U.n---> 1, u _L qn Vn, and qn ~ 0 weakly. We observe that the argument following (3.3) can be used to show that u is fixed by T k. Our goal is to reach a contradiction by showing that u is fixed by all the other k- 1 maps. If u is not fixed, note the first letters T n in FnT k (succeeding Tk) for which u is not a fixed point. Denote by A n the interceding words (which fix u). (In the event the word A n is null, replace it with the identity map I.) Since we are dealing with just k-1 distinct maps, we may assume by dropping to a subsequence that FnTkB n = ("') TAnTkBn, for somefixed map T e {T 1, ..., Tk_l}. Again in a manner similar to (3.3), we compute both (3.10) IITkBnVnll2- [IAnTkBnVnll 2 and (3.11) [[AnTkBnVnl[ 2 - IlTAnTkBnVnll 2. Now (3.10) = II~.nu + Tkqnll 2 - II%u + AnTkqnll 2 = (3.12) IlWkqnll2 - IIAnTkqnll 2 + (3.13) (~n u, Tkqn) + (Tkq n, 0tnU) (3.14) (anu, AnTkq n) - (3.15) (AnTkq n, OtnU). Now (3.13) ~ 0. Since the A n fix u, the An* also fix u, hence (3.14) = (C~nU, AnTkqn) = (CtnU, Tkq n) ~ 0 (and (3.15) ---> 0). The assumption (3.7) implies that (3.10) --->0. Hence [ITkqnll - I[AnTkqnl[ ---> 0. Since the A n ~ S(T 1 ..... Tk.1) and Tkq n --~ 0 weakly, we conclude by induction that AnTkq n --->0 weakly. This result enables us to compute (3.11) = [[OCnU + mnTkqn/t 2 - II0~nTu+ TAnTkqn[[ 2 = (3.16) II0qaull2- II0tnTull2 + (3.17) IlAnTkqnll 2- IlTAnTkqnll 2 + (3.18) (AnTkq n, r u) + (OtnU, AnTkqn) - (cr~Tu, TAnTkqn) - (TAnTkqn, ~nTu). Now (3.18) ~ 0 by AnTkq n ~ 0 weakly. As (3.11) --~ 0, we must have (as 3.17 is nonnegative) that Ilull2 - IlTull2 = 0. This implies u is fixed by T, as T has (W). This contradicts the construction of T. We conclude that Wnv n ---) 0 weakly, completing the proof. THEOREM. Let {T1, T 2 .... } be a set of contractions on a Hilbert space H, such that the algebraic semigroup S = S(T 1, T 2 .... ) has (USC). Let r be a mapping from the natural numbers N into itself, with the property that each range value is assumed infinitely often, and let S n = Tr(n) Tr(n_l) ... Tr(1) , and NQ = c3 n=l ~176{x [ Tr(n) x = x }. Suppose that for each fbced u, Ilull = 1, and only those n for which u is not a fuced point of Tr(n), 160 Dye

(*) SUpn IlWr(n)ull < 1. Then S n converges weakly to the projection Q on the subspace NQ. Proof. Because Tr(n) and Tr(n)* share the same fixed points, we see that Tr(n) Q = Q = Tr(n)*Q, from which it follows that Tr(n)Q = Q = QTr(n) for all n _> 1. Thus SnQ = Q, for all n. Note that the set { Tr(n)( I - Q ) } satisfies condition (*) if {Tr(n) } does. Hence to complete the proof, it will suffice to replace ( I - Q )H by H, assume NQ = (0), and show that S n converges weakly to zero. If Snx fails to converge weakly to zero for some x, we apply Lemma 1 to get a subsequence Snx = C~nU + v n where u ~ O, COn---)1, u I v n V n, and v n --4 0 weakly. Note the first letters T n in the original sequence succeeding S n (in the subsequence) for which u is not a fixed point. These always exist as NQ --- (0), and each T n appears infinitely often. Denote by W n the intervening words (which fix u). We compute IlWnSnx[I 2 = II%ull2 + IIWnvnll 2 + (3.19) (CXnU, Wnvn) + (Wnvn, t~.nU). Now (ctnu, Wnvn) --- (OtnU, Vn) ~ 0 as u is fixed by the Wn, hence also by Wn*. So (3.19) ~ 0. ]lSnxll 2 and IlWnSnXll 2 must share the same (nonzero) limit L, implying, after expanding IlSnxll2, that I[vn[I - IlWnvnll ~ 0. Hence Wnvn--+ 0 weakly, by (USC). We now compute (3.20) IITnWnSnxll 2 = (3.21) IlanWnu[I2 + (3.22) [ITnWnvnl] 2 + (3.23) (OtnU, Tn*TnWnvn) + (3.24) (Tn*TnWnvn, OtnU). By (USC), (3.23) --~ 0 and (3.24) ~ 0, as Wnv n ---) 0 weakly. Now condition (*) implies that SUPn IITnull _< ( 1 - ~ ) Ilull, for some 8 > 0. Hence lira SUPn (3.21) < Ilull 2, and as IITnWnvnll2 ___ Ilvnll2, we note that lim n IITnWnSnxll 2 < Ilull2 + limnl[Vnll 2 = lira n IISnxll2, a conta'adiction, as these two limits must both equal L. Hence Snx must converge weakly to zero. It is clear that afinite set of (W) contractions satisfy hypothesis (*) of the Theorem. In light of Proposition 2, we have COROLLARY. (Amemiya-Ando [1]). Let {T 1, T 2 .... TN} be afinite set of contractions on a Hilbert space H, each having (W). Let r be a mapping from the natural numbers N into itself, with the property that each range value is assumed infinitely often, and let S n = Tr(n) Tr(n_l) ... Tr(1), and NQ = n k= 1N{x I Tk x = x }. Then S n converges weakly to the projection Q on the subspace NQ. We finish by showing that the conditions (W) and (CN) have little to do with the Dye 161 question of strong convergence. Example. Let x 0, Xl, x 2 .... be an orthonormal base for the separable infinite dimensional Hilbert space H, and let U+ be the unilateral shift U+x n = Xn+ 1 where n ...... 0,1, Then U+ x 0 0 and U+* x n=xn_ 1 forn>2. As is easy to see,(U+) n converges weakly but not strongly to 0, whereas U+ *n converges strongly (and thus weakly) to 0. One has IIW+xll -- tlxll, for all x, but U+ ~ I. Thus, U+ does not satisfy (W). Neither does U+ , since the set of contractions satisfying (W) is closed under * (See [1]). However, the weak convergence could be predicted because U+ has (CN). Now let 0 < ~n < 1, ~n $1, and define Vx n = ~nXn+ 1, for n = 1, 2 .... (V is called a weighted shift.) It follows that IlVxll < Ilxll, if and only ifx r 0, so that V satisfies condition (W). In addition, Vmxn = ~n "'" 13n+m-lXn+m" Does {V m} converge strongly? Because V has no fix vectors ~ 0, one can show, V m converges strongly if and only if it converges strongly to 0. Moreover, as a simple approximation argument shows, Vmx --~ 0 for each x if and only if Vmxn --~ 0, for each n. Write [3n = 1 - Pn, where 0 < Pn < 1 and Pn $ 0. Thus we have the following: {V m} is strongly convergent if and only if, for each n, lira m --~ o, l-[k=n m (1 - Pk) = 0, or what is the same in view of conditions on the Pn, if and only if the infinite product I-Ik=i ~176(1 - Pk) diverges, namely, if and only if Y~k=l ~176Pk diverges. So, study of the strong convergence of {V m} reduces to study of the divergence of certain real infinite products. For stronger conditions on a contraction T so that its iterates converge strongly, see Amemiya and Ando [1] and Halperin [4]. ACKNOWLEDGEMENT: The author is particularly grateful to the referee for his many helpful corrections and suggestions on a preliminary draft of this paper.

REFERENCES

1. I. Amemiya and T. Ando, Convergence of random products of contractions in Hilbert space, Acta Sci. Math. (Szeged), 26 (1965), 239-244.

2. F. Browder, On some approximation methods for solutions of the Dirichlet problem for linear elliptic equations of arbitrary order, J. of Math. and Mech., 7 (1958), 69-80.

3. R.S. Foguel, Powers of a contraction in Hilbert space, Pacific J. Math., 13 (1963), 551-562.

4. I. Halperin, The product of projection operators, Acta Sci. Math. (Szeged), 23 (1962), 96-99.

5. B. Sz.-Nagy and C. Foias, Sur les contractions de respace de Hilbert. IV, Acta Sci. Math. 162 Dye

(Szeged), 21 (1960), 251-259.

6. John von Neumann, On Rings of Operators. Reduction Theory, Annals of Math., 50 (1949), 401-485.

7. M. Prager, On a principle of convergence in a I-filbert space, (in Russian), Czech. Math. J., 85 (1960), 271-282.

8. F. Riesz and B. Sz.-Nagy, Functional analysis, (translated by L. Boron from second French edition), F. Ungar, New York, 1955.

9. K. Smith, D. Solomon, and S. Wagner, Reconstructing objects from radiographs, Bull. Amer. Math. Soc., 83 (1977), 1227-1270.

Dept. of Mathematics, Cal State Northridge, 18I 11 Nordhoff St., Northridge, CA 91330 USA

Submitted: April 18, 1988 Revised: August 29, 1988 Integral Equations 0378-620X/89/020163-4851.50+0.20/0 and Operator Theory (c) 1989 Birkh~user Verlag, Basel Vol. 12 (1989)

THE FEYI~-KAC FORMULA WITH A LF~ESGUE-STIELTJES MEASURE: AN INTEGRAL EQUATION IN THE GENERAL CASE

Michel L. LAPIDUS 1

Let u(t) be the operator associated bypath integration with the Feynman-gac functional in which the time integration is performed with respect to an arbitrary Betel measure ~ instead of ordinary Lebesgue measure ~ . We show that u(t) , considered as a function of time t , satiafies a Volterra-StYeitjes integral equationj denoted by (~) . Ye refer to this result as the "Feynmau-Zac formula with a Lebesgue-Stieltjes measure 'r. Zndeedj when ~ = ~ , we recover the classical Feynman-gac formula since (~) then yields the heat (resp., Schrb~inger) equation in the diffusion (resp., quantum mechanical) case. We stress that the measure ~ is in general the sum og an absolutely continuous, a singular contJnuous and a (countably supported) discrete part. We also study various properties og (e) and of its solution. These results extend and use previous work of the author dealing with measures having finitely supported discrete part (Stud. Appl. Math. 76 (1987), 93-132)7 they seem to be new in the diYfusion (or '~maginary time'? as well as in the quantum mechanical (or "real time'? case.

1. INTRODUCTION

The purpose of this paper is to establish an intriguing extension of the classical Feynman-Kac formula: namely, the "Feynman-Kac formula with a (general) Lebesgue-Stieltjes measure". In a previous work [18], we have obtained this formula for a measure with finitely supported discrete part (Y.e~ a measure which charges a finite number of points). Our present aim is to extend this result to an arbitrary

IResearch partially supported by the National Science Foundation under Grant DMS 8703138. This work was also supported in part by NSF Grant 8120790 at the Mathematical Sciences Research Institute in Berkeley, U.S.A., the CNPq and the Organization of Latin American States at the Institute de Natem~tioa Pure E Aplicada (IMPA) in Rio de Janeiro, Brazil, as well as the Universit5 Pierre etNarie Curie (Paris VI) and the Universit~Paris Dauphine in Paris, France. 164 Lapidus

Borel measure and to understand it from the point of view of integral equations. After having reformulated the main result of [18], we approximate a general Borel measure by a sequence of measures with finitely supported discrete part in order to derive a Volterra-Stieltjes integral equation valid in the general case. We now recall some of the main facts pertaining to this subject. In the early 1950's, Kac [15], motivated by the beautiful work of Feynman [6] in the context of quantum mechanics on what came to be known as "Feynman path integrals", used the well defined notion of Wiener's probability measure introduced in the early 1920's [28, Part IC] in order to deal with the diffusion Or "imaginary time" case. In this way, he was able to express the solution of the heat equation, or of an equivalent integral equation, in terms of a bona fYdepath integral. The reader should be aware of the fact that the "Feynman path integral" is not an integral in the mathematical sense of the term. The work of Feynman, Kac and their followers gave rise to a flourishing literature and developed into a new subject called "functional integration". The use of functional (or path) integrals has given rise to many interesting results in pure and applied mathematics [16]. Recently, it led to an insightful new proof of the famous Atiyah-Singer index theorem. (See [8] and the references therein.) It permeates, often in heuristic form, much of modern theoretical physics, including statistical and quantum mechanics [16,25], quantum field theories [9], gauge theories [5, w and, more recently, the very active area of string theory (see, e.g., [30]). Actually, we believe that the present work, together with its companions [17,18] and the related works by Johnson and the author [12,13,14] on Feynman's operational calculus for noncommuting operators [7], as well as [19,22], may be of some relevance to aspects of the latter subject. We now briefly describe the main result of this paper. Most of the terms used below will be precisely defined in the next section. Given t ~(a,b] , we consider the following "functional" on Wiener space:

F(x) = exp(/[a,t) 0(s,x(s)) n(ds)) , (i.I) associated with the "potential" 0 and the Lebesgue-Stieltjes measure defined on the time interval [a,b] . Let u(t) be the corresponding operator obtained by path integration (diffusion case: ~ = I and 0 = -V) , Lapidus 165

followed by analytic continuation and continuity in a complex parameter (quantum mechanical case: ~ = -i and e = -iV) . It is shown by Johnson and the author in [13] that this procedure is well defined and that for a fixed time t , u(t) can be expressed as a time-ordered perturbation expansion or "generalized Dyson series". We establish here that u = u(t) , considered as a function of time, satisfies the following Volterra-Stieltjes integral equation on [a,b] (see Theorem 4.37: -(t-a)(Hol~) -(t-s)(Hol~) u(t) = e + /[a,t) e O@,n(s) u(s) q(ds) , (1.2) where -H 0 is the Laplacian or "free Hamiltonian" in L2(; N) and O@,~(s) := @(n({sJ)O(s)) 0(s) , with @(z) := (e z - l)/z and @(0) = I. Note that when s is not a "pure point" of n (Y.e., when n({s}) = O) , we have 0@,n(s) = O(s) , the multiplication operator by B(s,.) in L2(l N) . [We assume in this paper that 8(.,.) : [a,b] • E N ~ ~ is such that /[a,b) [[0(s,-)]]m [~[(ds) < +m ; this is the case, for instance, if O is measurable and bounded.] In the classical case where n = ~ := ordinary Lebesgue measure on [a,b] , (1.2) reduces to the usual integral equation, which yields the heat (resp., Schr~dinger) equation with potential V in the diffusion (resp., quantum mechanical) case. This is why we refer to our result as the "Feynman-Kac formula with a Lebesgue-Stieltjes measure". The latter seems to be new in both the diffusion (or "imaginary time") and the quantum mechanical (or "real time") cases. We stress that the original Feynman-Kac procedure consisted in deriving a path integral associated with the heat (or SchrSdinger) equation whereas we start here from a path integral [associated with the functional in (i.I)] and then obtain a corresponding integral equation. We also emphasize that in our present setting, q is an arbitrary Borei measure on [a,b] . For example, it may very well have a discrete part; moreover, its continuous part is in general the sum of an absolutely continuous measure and of a singular measure. Actually, the reason for considering here an integral rather than a differential equation is that n may have a nontrivial singular part. The case, studied in [18], of a measure with finitely supported discrete part, is probably the most physically relevant. (See however Section 5.B.IV.) We mention that physical interpretations are discussed in both the 166 Lapidus

quantum mechanical and diffusion cases in [18], especially in Section 6.C; further, some of the results of [18], combined with those of [13], have led to a new noncommutative multiplication of Wiener functionals and to a deeper understanding of Feynman's operational calculus (see [14]). From a mathematical point of view, however, it is quite natural to consider the general case; furthermore, our present study also sheds new light on the work in [18] and [13]. (See in particular Theorems 3.2 and 5,1, as well as Remarks 4.3 and 5.1.) Throughout the text, we point out connections between our results and some extension of work of Dollard and Friedman [3; 4, Chapter V] on product integrals of measures in a finite dimensional context. (See Remark 4.3.) This topic is pursued further in a forthcoming paper [20]. Without going into the details here, we mention that these connections can be established only after our main result, the integral equation of Theorem 4.3, has been obtained. We now indicate how the rest of this paper is organized. In Section 2, we present notation and preliminary results. In Section 3, we consider the case of a measure with finitely supported discrete part and reformulate the integral equation obtained in [18]. In Section 4, we treat the case of an arbitrary Borel measure. By means of the "generalized Dyson series" derived in [13], we establish an approximation theorem for u(t) by operators Um(t) associated with measures with finitely supported discrete part (Section 4.B) and then use the results of Section 3 to derive the Volterra-Stieltjes integral equation in the general case (Section 4.C). In Section 5, we show that the latter integral equation has a unique bounded solution, necessarily equal to u(t) [Section 5.A], and that u(.) is discontinuous at every "pure point" of ~ (Section 5.D); we also discuss various special cases of the equation (Section 5.B). Finally, in Section 6, we examine the propagator associated with the integral equation and give a sufficient condition under which it is a unitary operator in the quantum mechanical case. A preliminary version of this paper is included in the preprint series of the Mathematical Sciences Research Institute in Berkeley (MSRI 02611-87). This work was presented in part at the "u Latin American School of Mathematics," held in July 1986 at IMPA in Rio de Janeiro and - in its final form - at the conference on "The Path Integral Method and its Applications," held in September 1987 at the International Centre for Theoretical Physics in Trieste. Lapidus 167

2. NOTATION ANDPRELIMINARIES

2 .A. Notation

Let N be a positive integer and let a, b be real numbers such that a < b . Let M([a,b]) be the space of complex Borel measures on [a,b]. Given ~ ~ M([a,b]) , we say that a complex-valued Borel measurable function 8 on [a,b] x I n belongs to Lml;D = Lml;~ ([a,b)) if

[l~ := I[a,b) [[O(s,-)[[o [nl(ds) < +~ , (2.1) where In[ is the total variation measure of n When this is the case, it follows in particular that []8(s,')[[~ is finite and 8(s) ~ L(L2(IN)) for [n]-a.e. (almost every) s in [a,b) . Here we denote by %(s) the operator of multiplication by the function 8(s,.) in L2(IN) and by L(L2(IN)) the space of bounded linear operators from L2(I N) into itself; note that 8(s) has norm []8(s)[[ = []8(s,.)[], . Recall that any D ~ M([a,b]) has a unique decomposition in the form n = p + v , where p ~ M([a,b]) is continuou; (i.e., p({s}) = 0 for every s in [a,b]) and ~ ~ M([a,b]) is discrete (i.e., v = E~p=1 ~p 8 P where {Up} is a summable sequence of complex numbers {ip} C [a,b] , and $ is the Dirac measure of mass one concentrated at Ip) ; see [l; 24, p. 22]. -zH 0 We denote by e the analytic semigroup generated by the (normalized negative) Laplacian H 0 = - ENa=I 82]@x~ in L2(iN) ; -zH 0 recall that l[e ]] S 1 for Re z ~ 0 . Let ~, C+ and C~ denote respectively the complex numbers, the complex numbers with positive real part and the nonzero complex numbers with nonnegative real part. Let i = 4-i Let C = 6~[a,b]) be the space of continuous functions from [a,b] to i N . The Wiener space CO = C0([a,b]) is the set of all x ~ C such that x(a) = 0 , equipped with Wiener measure m [10,29,31]. Fix t ~ (a,b] . Given a function F: C~ ~ , * ~ L2(| N) , G i N and I > 0 , we consider the expression 168 Lapidus

(K~(F)$)(~) = fC0 F(k-I/2x + $) #(k-I/2x(t) + $) m(dx) . (2.2)

The operator-valued function space integral K~(F) exists for k > 0 if (2.2) defines K~(F) as an element of L(L2([N)) . If, in addition, K[(F) , as a function of k , has an extension to an analytic function on ~+ and a strongly continuous function on ~+ , we say that K~(F) exists for all ~ 6 ~ When k is purely imaginary, K~(F) is called the (analytic) operator-valued Feynman integral of the "functional" F . (See [13, Definition 0.I, p. I0].) In [13, Theorem 2.1, pp. 27-29 and Corollary 2.1, pp. 33-34], Johnson and the author show that for a very large class of functionals including that studied below, K~(F) exists for all ~ ~ ~ and is given by a time-ordered perturbation expansion, called a "generalized Dyson series". As is articulated in [13], this generalized Dyson series provides a way of carrying out the "disentangling process" that is central to Feynman's operational calculus [7, p, Ii0]. Given ~ ~ M([a,b]) and B ~ L l;q([a,b)) , we consider the exponential functional

F(x) = exp(f[a,t) 0(s,x(s)) q(ds)) , x fi C , (2.3) associated with the "potential" 8 and the "Lebesgue-Stieltjes measure" For natural physical and mathematical reasons [18, Remark 3.5, p. II0], we shall work with Kt(F) * , the Banach adjoint of K~(F) , rather than K~(F) itself; (of course,"L2(~ N) is here identified with its own dual. ) Set u(t) = K~(F)* , (2.4) for t ~ (a,b] and ~ ~ ~+ ; we let u(a) = I, the identity operator. As in [18], we refer to the case when ~ = -i and 8 = -iV (reap. , k = I and 8 = -V) with V ~ L l;q , as the quantum mechanical (reap., probabilistic or diffusion) case. (See especially [18, w pp. 127-129].)

Remarks 2.1. (a) The function F in (2.3) is not necessarily defined on all of C . However, it is shown in [13, Lemma 0.I, p. 12] that Lapidus 169 for every k > 0 , F(k-I/2x + ~) is defined for m • Lebesgue-a.e. (x,~) ~ CO • I N The corresponding measure theoretic problems are dealt with in [13], especially pp. 10-12, 33-34 and 69-70. (b) The operator u(t) in (2.4) also depends on k ~ ~ ; for simplicity, however, we do not indicate it explicitly. (c) We choose the Banach adjoint instead of the Hilbert adjoint in (2.4) as a matter of convenience (see [18, Remark 3.5, p. II0]). For some purposes, the Hilhert adjoint might be better suited. (d) Most of the results of this paper could be formulated on the interval [a,+~) rather than on [a,b] by considering set functions on [a,+e) whose restriction to [a,b] lies in M([a,b]) for any b > a . This will be briefly indicated in Section 6. Furthermore, the replacement in (2.3) of the interval of integration [a,t) by (a,t] would lead to minor technical changes throughout the paper. We note that the measure D is allowed to have "mass" at the endpoint a or at b ; the latter fact will be useful in some examples (see the Trotter products of Section 5.B.III) while the former will enable us to obtain a simpler formulation of the integral equation (see Sections III and IV.C).

For a more detailed discussion of some of the above notation, the reader may find it useful to consult [13, w pp. 6-14] and [18]. For a concise exposition of Bochner integration theory, we refer to [11, Chapter III]; for the theory of Brownian motion and the applications of path integrals, we mention [9,10,16,25,29,31]; finally, the basic elements of measure theory used in this paper can be found in [1,27] and [24, pp. 12-26].

2.B. PreliminarYes

Given n ~ M([a,b]) and e g L l;n([a,b)) , we set for s ~ [a,b)

0 ,n(s) = )(n(s)0(s)) 0(s) , (2.5) where ~(s) := n({s}) and ~ is the entire analytic function defined by n-i ~(z) = ~ n=l zn! , for z ~ ~ ; that is,

Z

~(z) = ? - 1 ) for z # 0 , and ~(0) = 1 (2.6) g 170 Lapidus

Note that 0 is defined by means of the functional calculus in the Banach

algebra L(L2(RN)) . Strictly speaking, since 8 ~ L l;q and by (2.1),

e q(s) is only defined for lq[-a.e, s in [a,b) ; in particular, e ,q(s) is well defined for every "pure point" (or "atom") s in [a,b) (that is, every s such that q(s) ~ O) and for Ipl-a.e. s in [a,b) , where p denotes the continuous part of q .

Remarks 2.2. (a) A similar notion, but in a somewhat different context ahd with different notation, is used in [3; 4, Chapter 5]. 9 (b) Note that 8 is the multiplication operator in L(L~(RN)) ~,q by the Borel measurable function, still denoted 8 ,q , defined as follows:

8 ,q(s,~) : ~(q(s)e(s,()) e(s,~) , for (s,() ~ [a,b] x ;N . (2.7)

In the course of this paper, we shall need the following simple result (compare [4, pp. 159-161]).

LEI@IA 2.1. (a) If s is noC a pure point of q (i.e., q(s) = 0),

then o ,n(s) = O(s) (2.8)

(b) If s Js a pure point of q , then

q(s)e n(s) = e q(s)o(s) - I 2,9)

and IIq(s)%,q(s)ll ~ e In(s)llle(s)ll 1 . 2.10)

(c) Under the above hypotheses, O, n fi L~l;q and

11e I1~ol ; n Ileq~, qll'~l ;n Se -I . 2.11)

PROOF. (a) If q(s) = 0 and O(s) lies in L(L2(~N)) , then by (2.5), O ,q(s) = ~(0) 0(s) = O(s) since 4(0) = I by (2.6) . Note that, strictly speaking, (2.8) only holds for [pl-a.e. (non-pure point) s in [a,b]. Lapidus 171

(b) If s is a pure point of q , then 8(s) belongs to L(L2(IN)) and by (2.5) and (2.6),

n(s)O ,~(s) = ~(n(s)O(s)) n(s)e(s)

= e n(s)O(s) - I ; (2.12) here we use the fact that, according to (2.6), we have

A~(A) = ~(A)A = e A I , for any A ~ L(L2(~N))

This establishes (2.9). Now (2.10) follows from (2.12) and the inequality

n,,eA - zll -< I: = e tIAll - 1 , for A ~ L(L2,=,/twN~ . n=t

(c) As was observed in Remark 2.1(b), the function 0 ,q: [a,b] • R N + ~ is Borel measurable and ll0~,q(s,')]Im = l]0~,q(s)ll for lq[-a.e, s in [a,b) Now, by (2.5),

llo~,n(s)ll ~ ~(lloll~l;n) llo(s)ll for Inl-a.e. s ~ [a,b) (2.13) and hence

II%,nll~l;n = f/a,b) II~ < ~(ll~ [f[a,b) IlO(s)lllnl(ds)] II o IIo~i; ,~ = qD(Uoll~,l; n) lloIl,ol;n =e -I<+~.

Thus 0 ,~ ~ Lml;q and (2.11) is proved. Note that, in view of (2.5), (2.13) follows from the fact that for Inl-a.e. s ~ [a,b) , ~(s)0(s)l] []011ml;~ , and the following simple observation:

I]~(A)[[ ! ~(a) if [JAIl S a , for A ~ L(L2(|N)) ; indeed, m An-I m llAlln-1 lh>(A)ll = U Z --7T-., I[ _< ~ n] = ~(IIAII) -<" qD(a) . n=l ' n=l

This concludes the proof of Lemma 2.1. o 172 Lapidus

3. MEASDREWITH FINITELY SUPPORTED DISCRETE PART

In this section, we reformulate the integral equation obtained by the author in [18] in the case of a measure with finitely supported discrete part. This will serve as the basis of our study of the case of a general measure in Section 4. Let n = p + be the unique decomposition of the Borel measure B into its continuous part and its discrete part ~ . Throughout this section, we assume that has a finite number of pure points, i.e., that the support of v is finite. We may thus write

m

p=l P Zp where m ~ ~ for i ! p E m and P

t 0 := a ~ ~I <"'< ~m <- b = : ~m+l " (3.1)

Remarks 3.1. (a) In the following, we set ~0 = 0 if t 0 := a < ~I and mO = ml if t0 = ~I (see [18, Remark 2.1, p. 98]); note that according to this convention, we always have w 0 = ~(a) (b) In [18], we have used h instead of m in the analogue of (3.1).

We now recall the main result obtained by the author in his previous work focusing on the case of a measure with finitely supported discrete part [18, Theorem 2.1, p. 98, as well as w and w

TKEOREM 3.1. Let u(t) = K~(F)* , where F is siren by (2.3) and n is defined as above. Then for each fixed p ~ {0,...,m} and for a77 t ~ (r ] , the operator-valuedfunction u satisfies the fo]]owinj Vo]terra-Stie]tjes integra] equation:

-(t-~p)(Ho/k ) ~pS(Ip) t -(t-s)(HO/k) u(t) = e e U(~p) + f~ e 0(s)u(s)p(ds) . (3.2) P The gochner integral in (3,2) converges Yn the strong operator topology, ~oreover, for all t @ [a,b] ,

l[u(t)l] <- exp(f[a,t ) li0(s)II l~l(ds)) <_ ~xp(H01~1;n) Lapidus 173

Further, u is the unique bounded operator-valued function on [a,b] that satisfies (3.2) on (xp,Xp+ I] for p = 0 .... ,m .

Although the above statement is rather concrete and easy to apprehend, we need to reformulate it in a form that is better suited to our purpose~ in particular, we obtain a single integral equation written in a somewhat more invariant form.

T~OREM 3.2. For all t ~ [a,b] , the operator-valued function satisfies the follovingintegra2 equation: -(t-a)(Ho/t) -(t-s)(Ho/l) u(t) = e + f[a,t) e 8 ~q(s) u(s) ,(ds) . (3.3)

PROOF. Fix t ~ [a,b]. We shall show that u(t) is given by the right-hand side of (3.3). If t = a , this is obvious since u(a) = I We now assume that t ~ (a,b]; let p be the unique integer in {0, .... m} such that t ~ (~p,~p+l ] . According to Theorem 3.1,

u(t) -(t-Xp)(H0/k) e mpS(Ip) - e U(Xp) -(t-s)(HO/X) + /~ e O(s) u(s) p(ds) P Hence, since B is continuous and since in short-hand notation,

t

q=a ~q-I p where ~ = 1 if q(a) = 0 and a = 2 if ~(a) # 0 (recall that tO = Xl = a if D(a) # O) , we have: -(t-tp)(Ho]~) e~pS(tp) u(t) = e U(~p)

-(t-s)(Ho/l) + f~ e e(s) u(s) ~(ds)

p t -(t-s)(KO/l) - Z Ix s e 8(s) U(s)~(ds) . q=a q-I

By inserting a telescoping sum, it comes:

u(t) = e-(t-a)(Ho/l)e q(a)e(a) -(t-s)(Ho/%) + f~ e e(s) u(s) ~(ds) 174 Lapidus

P -(t-Tq_l)(H01k) emq_10(Tq_l ) [e U(Tq_l) q=o~ -(t-s)(H0/k) + /~q-lq e 0(s) u(s) ~(ds)]

P -(t-~q)(H0/k) e~qO(~q) + ~ e U(~q) . (3.4) q=~

Here we used the convention specified in Remark 3.1(a) and the fact that u(~ 0) = u(a) = I to deduce that

-(t-T~_I)(H0/k) ~_IS(T~_I ) -(t-a)(H01k) en(a)0(a ) e e u(T_l) = e

Now, by Theorem 3.1 - more precisely, by Eq. (3.2) applied to ~q E (~q_l,~q] for q E {1,...,m} - we see that

-(~q-lq_l)(Ho/k) e~q_10(~q_l ) U(Tq) = e U(~q_ I )

e -(~q-s)(HO/~) 0(s) u(s) ~(ds) . (3.5) + f~-I

Thus, by [II, Theorem 3.7.12, p. 83] and by the semigroup property of -P(H01k) {e }p>0 ' we have in particular for q ~ {l,...,p}:

-(t-Tq)(H0/l) e U(Tq) -(t-Tq_l)(Ho/~) e~q_iO(~q_l ) = e U(~q_ I ) -(t-s)(H0/~) + f~]-i e 8(s) u(s) p(ds) . (3,6)

Combining (3.4) and (3.6), we obtain:

-(t-a)(H0/k) eq(a)0(a ) + ~ e -(t-s)(H0/%) u(t) = e 8(s) u(s) p(ds)

P -(t-~q)(Ho/k) [emqfl(Tq) + E e I] U(~q) (3.7) q=~ Lapidus 175

By Lemma 2.1(a),

0 ,~(s) = e(s) , Ipl-a.e. in [a,t) , (3.8) and since for q ~ {0,...,m} , ~(~q) = ~q , it follows from (2.9) that

~q e D(~ q) = e - I (3.9) and hence in particular -(t-a) (H0/X) e en(a)e(a) -(t-a) (No/k) -(t-a)(H0/%) = e + e 8 ,~](~0)~ 0 .

Consequently, in view of Remark 3.1(a), (3.7) becomes successively:

-(t-a)(Ho/~) -(t-s)(Ho/~) u(t) = e + [[a,t) e e ,n(s) u(s) p(ds)

P -(t-~q) (Ho/~) + ~ e 0 ,n(~ q) U(~q) ~q q=1

-(t-a)(H0/~) -(t-s)(Ho/%) = e + f[a,t) e e n(s) u(s) p(ds)

-(t-s)(Ho/~) + f[a,t) e e ,~(s) u(s) v(ds)

-(t-a)(H0/%) -(t-s)(H0/%) (s) u(s) n(ds) = e + f[a,t) e e ,n

This proves (3.3), as desired.

4. GENERAL BOREL MEASURE

Let ~ be an arbitrary complex Borel measure on [a,bl Let

= p + v (4.1a) be the unique decomposition of D into its continuous part p and its discrete part v . We write 176 Lapidus

= 2 ~ 8 , (4.1b) p=l P ~p where I, }~ C [a,b], {~p}p=l~ C ~ and ~ ] < +~ . Note that the ~ p=l p=l[~P 's, assumed to be distinct, need not be naturally ordered. P Let u(t) = K~(F) ~, where F is given by (2.3) . After having written a perturbation expansion for u(t) in Section 4.A, we shall establish an approximation theorem by measures with finitely supported discrete part in Section 4.B and then use the results of Section 3 in order to obtain an integral equation valid in the general case, in Section 4.C.

4.A. 6eneraiizedDyson Series

Let t ~ (a,b]. It is convenient to introduce the following

notation; for p _> i, we set if ~ ~ [a,t) (4.2) otherwise.

So that the restriction of $ to [a,t) is equal to Zm p=l ~p,t6~ P

In view of [ii, Theorem 3.8.1, p. 85], which enables us to take the adjoint and reverse the time-ordering [18, Remark 3.5, p.llO], the next result is a direct consequence of [13, Corollary 2.1, pp. 33-34 and Example 3.4, pp. 40-41]. (Note that in [13] all the theorems are stated for a fixed time t .) For simplicity of notation, we assume that a is not a pure point of ~ ; see Remark 4.1(d) below for the general case.

TI{EOR~ 4.1. 6iven t 6 (a,b], Jet F be defined as in (2.3) with q as in (4.1) and q(a) = 0. Then for a]l k ~ ~+, u(t) = K~(F)* is given by the following "generalized Dyson series" or "time-ordered e.~Tonential series ":

u(t) = l l n=0 h=O q0+ql+...+qh=n,qh # 0 Lapidus 177

ql ql (~l,t) ... (~h,t) 2 ql ! ... qh ! j i +. .. +jh+Jh+l=qO

u . (t) , (4.3) n,qo;Jl,...,Jh+ I

~here for nonnegative integers qo' ql""'qh and jl,..,jh+ 1 such that qh # O, qo +'"+ qh = n and Jl +'"+ Jh+l = qo '

Un,qo;Jl,...,Jh+l(t)

= (t) ~,qo;jl, (s.,..., ;t) /Aqo;Jl .... 'Jh+l .... 3h+l • Sqo

(~•215 (4.4)

Here, for each fixed h , o denotes the permutation of {i,.. ,h} such that

~(i) < "" < ~(h) (4.5) and ) ~ (a,t)qO : Aq0;j I .... ,Jh+l (t) = ~(s I ..... Sq0

a < s I < ....< s31 < %a(1) < Sjl+l < "'" < Sjl+J2 < ~(2)

.. < < sj <...< s < t~ ; (4.6) < Sjl+J2+l < < Sjl+...+j h %~(h) l+...+Jh+l qo further, for (Sl,..., Sq0) ~ Aqo;Jl,...,Jh+l, (t) ,

Ln,q0;Jl,...,Jh+l(S 1 ..... Sq0 ;t)

-(t-Sq0)(Ho/l) -(Sqo - Sqo_l)(H0/l) = e 8(Sqo) e 178 Lapidus

e(Sqo_ 1) ... 0(Sjl+...+jh+ I)

-(Sjl+...+jh+ 1 ~o(h))(H0/k) e [0(~a(h))] qa(h)

-(Za(h) - sjl+...+jh)(Ho/l) e 0(Sjl+...+jh)

-(Sjl+J2+ I - ~ (2))(H0/X) .. 0(Sjl+J2+ I) e

qo(2) -(zo(2) - Sjl+J2)(H01k) [O(~a(2))] e

-(sjl+t - ~ (1))(Ho/k) , e(sj l+J2 ) "" 8(Sjl +1 ) e

q~(1) -(z~(1) - Sjl)(H01k) [0(~(i))] e 0(Sjl )

-(s2-sl)(H01i) -(sl-a) (H011) .. %(s2) e 8(Sl) e (4.7)

The sez~es in (4.3) is (abaoiutely} summable in L(L2(RN)) and the integral in (4.4) is a strong Boehner integral, Moreover,

Ilu(t)ll ~ exp(/[a,t) UO(s)lllnl(ds)) S exp(lletl~l;n) . (4.8)

Remarks 4.1. (a) Actually, L and n,qo;Jl,"',Jh+ 1 Un,qo;Jl,...,jh+ 1 also depend on qi ""qh '" for simplicity, however, we do not indicate it explicitly. Note that our notation deviates from that of [18, Theorem 4.1, pp, 110-112] and [!3, Example 3.4, pp. 140-41]. Lapidus 179

(b) For what follows, it is important to observe that , Aqo;j . and hence u are Ln,qo;Jl,...,Jh+ 1 1,...,Jh+l n,qo;Jl,...,Jh+ 1 independent on the weighting coefficients ~p,t s . (c) If the ~p'S can be ordered in such a way that {~p}~=l~ is a nondecreasing sequence, then the permutation ~ in (4.5) can be chosen equal to the identity [13, p. 41]; moreover, in this case, if p is such that t ~ (Ip,lp+l] , then u(t') is given by the same generalized Dyson series for all t' ~ (~p,~p+l ] . (d) If a is a pure point of I] (f.e., ~(a) # 0), then u(t) = u(t)e N(a)~(a), where u(t) is given by the right-hand side of (4.3) but corresponds to the restriction ~ of n to the interval (a,b]. (Note that Theorem 4.1 applies here since ,(a) = 0.) As is explained in [18, Remark 4.5, p. 118], this is a very special case of the phenomenon of "change of initial condition" and follows easily from (2.2) and (2.3) since, for example;

exp(/[a,t) 0(s,x(s))n(ds))

= exp(/[a,t) 0(s,x(s))~(ds)) exp(~(a)e(a,x(a))

4.B. Approximation by Heasures wi~h FiniteTy Supported Dfscrete Part

Given a positive integer m , we set

~m P + ~m ' (4.9a) where p is the continuous part of D and

m v = Z m 6 (4.9b) m p=l P *p

Let Um(t) = K~(Fm)*,_ where Fm is defined as in (2.3) except with replaced with Dm " We can now state our basic approximation theorem. 180 Lapidus

THEOREM 4.2. For all m ~ i,

sup []u(t) - Um(t)ll ! cm , (4.10) t~[a,b],k~

~here c m ~ 0 as m ~ ~ . (Hore precisely, Cm is given by Eq. (4.22) below.) fn partYcuJar, Um(t) ~ u(t) convergos in L(L2(IN)) as m ~ unifor~lyfor t G [a,b] and ~ G ~

PROOF. In light of Remark 4.1(d), it suffices to assume that n(a) = 0. (See Remark 4.2(a) below for the case when a is a pure point of q.) Observe that it then follows that nm(a) = 0 for every m 2 i. Fix k ~ ~ and an integer m ~ i. Rewrite Vm in (4.9b) as follows:

~o

m p=l P ~p where, for p ~ i ,

m' = I mp if 1 < p < m

P t 0 otherwise. (4.11)

Fix t ~ [a,b] In the following, ~p,t is defined as in (4.2) and m' is defined similarly except with m replaced with m' If p,t p p t = a, [[u(t) - Um(t)] [ =]]I - I]] = 0 ; so that we may assume that t ~ (a,b]. Now, Theorem 4.1 applied both to n and qm yields:

u(t) - Um(t) = E E Z n=O h=0 qo+ql+...+qh=n,qh#0

ql qh (ml,t)ql . (m~,t)qh] [(ml~t) .. (~h,t) .. [ ql' qh' - ql'" qh' J

u . (t) , (4.12) Z n,qo;Jl,...,Jh+l Jl+.-.+Jh+l=q0 where u (t) is given by (4.4) n,q0;Jl,...,Jh+ 1 Lapidus 181

Let n, h, qo' ql"'"qh be nonnegative integers such that qo+ql+...+qh = n with qh # O. According to (4.2) and (4.11), h ~ m+l implies that ~)h,t = O; since qh # O, it follows that (~,t)qh = 0 and

(~01, t) ql ... (~ ,t)qh = 0 for h ~ m + I . (4.13a)

On the other hand, it results from (4.2) and (4.11) that if h ~ m, then ~' for all q ~ {i, ,h}" hence q,t = mq,t "'" ' (~ ,t)ql ... (o~ ,t)qh = (~l,t) ql ... (~h,t) qh for h ~ m . (4.13b)

Consequently, (4.12) and (4.13) yield:

u(t) - Um(t) = E ~ n=O h=m+l qo+...+qh=n,qh#O

ql qh (~l~t) ... (~h~t) ' ~ u (t) . (4.14) ql r ... qh .! jl+,,.+Jh+l=qO n'qo;31"'"Jh+l' .

Now, we claim that h (t)[[ E N Ile(~a)/I qa /A (t) Ile(Sl)l/'"lle(Sqo)ll IlUn'qo;Ji ..... Jh+l a=l qo;Jl ..... Jh+l

(Iplx...• , (4.i5) where Aqo;j I ..... Jh+l(t) is given by (4.6) .

-n(Holk) In fact, by (4.7) and since lle ii ~ 1 for p > 0 and ~ ~ , we have for $ ~ L2(i N) and (Ipl• (s I .... ,Sqo ) in t) Aqo;Jl,...,Jh+l( :

- . ;t)$ H HL,qo;j 1 ..... Jh+l(Sl ''" 'Sqo

h iiqc(~ ) [l~llll~ "'" H~ ~=lI IIo(~o(~)) " 182 Lapidus

= II~lllle(sl)ll "'" Ile(Sqo)ll ~:lnh lie( ~)llqa (4.16) in the last line of (4.16), we have used the fact that o , which occurs in (4.5)-(4.7), is a permutation of {l,...,h} . In view of (4.4), (4.16) implies (4.15). It now follows from (4.14) and (4.15) that

]lu(t) - Um(t)ll i E E E n=O h=m+l qo+...+qh=n,qh#O

([al,tlH~ ''" (]ah,t II0(lh)II)qh A (4.17) ql ! qh ! ' where we have set

A = E IA Ilo(Sl)l] )lI Jl+..,+Jh+l=qO qo;Jl ..... ]h+l (t) "'" ]IO(Sqo

(Ivtx..• .....l, dSqo) , (4.18)

For a fixed h and hence for the corresponding permutation of {l,...,h} , one deduces from a sectioning argument (recall that ~ is a continuous measure) that

Aq0(t) := {(s I .... ,Sq0) g (a,t)q0: a < s I < ... < Sq0 < t}

(4.19) : U Aq0;j I ..... Jh+l(t), Jl+..-+Jh+l:q 0 up to a set of (Ipl•215 zero.

It thus follows from (4.18), (4.19) and the "simplex trick" (see [13, bottom of page 18]) that

A : IA qo (t)Ile(Sl)ll "'" HO(Sqo )ll (Iplx...xl~l)(ds 1, .... dSqo) Lapidus 183

= 1 (f(a t) Ue(s)lllul(ds))q0 < L'i (llellgol:u) q~ (.4.20) qo ! ' - qo" "

If we note that, by (4.2), [Up,t[ 5 [Up[ for all p 2 i, it follows from (4.17), (4.18) and (.4.20) that

sup Ilu(t) - Um(t)[] <- c m , (.4.21) t~[a,b],k~

where

OO O0 c := E E l: m n=O h--m+l qo + ql+...+qh=n,qh#O

(llen~l;~)qo ql (lullllO(Xl)ll) ... (lr qh (4.22) qo ! ql ! ... qh !

[We indicate in Remark 4.2(a) below how (4.22) should be modified in case n(a) # 0.] We now conclude the proof of Theorem 4.2 by showing that c ~ 0 m as m ~ go ; this is achieved by observing that c is the remainder term of an m absolutely summable series. Indeed, the "NO-nomial formula" [13, Eq. (3.23), p. 41] yields:

go go qo ql qh ( ~ bp) n = E nl bo bl .. b h p=O h=O qo+ql +. +qh=n, qh #0 qo ! ql ! "'" qh ! (4.23)

It thus follows from (4.23), with b 0 = ~0Ugol;p and bp = [UplUo(~p)ll for p ~ i, that

GO gO 1 n=O h=O qo+ql +. ..+qh=n,qh#O

h-L ([[0 q0 .. qh ! ~gol;p ) ([ul[[]e(T1)U) ql . (l~hlll0(~h)]]) qh qo ! ql ! ... 184 Lapidus

Q:) 1 7., tt~ + r I~plllOtXp)ll) n n=O p=l

II o I1<; + lloll~l;v) n n=O

co I n-$ II01f=l;n} n n=O

= exp(1]OIl~l;~l) < +m; (4.24)

here we have used (4.1) and [13, Eq. (O.tl), p. 9]. This shows that the sums in n and h can be interchanged in the first term of (4.24). In view of (4.22), this proves that c + 0 as m + ~. m Theorem 4.2 now fellows from (4.21). o

Remarks 4.2. (a) If n(a) # O, then according to Remark 4.1(d) and since D(a) = qm(a), we have for m ~ i:

u(t) - Um(t) = u(t)e n(a) O(a) Um(t)enm (a)O(a)

= (u(t) - Um(t))en(a)O(a) ,

where u (resp., Um) is associated with the restriction of ~ (resp., qm ) to (a,b]. Consequently, (4.21) remains valid if, in (4.22), the left-hand side is replaced by e[q(a)]l[O(a)IlCm and the right-hand side is interpreted as corresponding to the restriction measures. (b) In [13, Example 4.3, pp. 59-61], is stated an approximation theorem similar to Theorem 4.2. However, since the former is obtained by means of the dominated convergence theorem for functional integrals, it only yields an estimate of the type (4.10) for ~ > 0 and no~ for % purely imaginary. It would still be possible to derive Theorem 4.3 below from this result but this would require, after having established the integral equation for % > 0 (and not directly for % ~ ~ as will be done below), two additional steps: analytic continuation (~ ~ 6+) followed by passage to the limit (X ~ ~) Lapidus 185

Note that it is the use of the generalized Dyson series obtained in [13] that enabled us to establish estimate (4.]0) uniformly in k [ ~ . (Compare the discussion in [13, top of p. 57].)

4.C. The ~Dlterra-Stieltjes InteEral Equation

We can now establish the main result of this paper. Following our previous terminology []7,18], we shall refer to it as the general "feynman-Kac formula with a s measure".

THEOREM 4.3. Zet ~ ~ ~; Let ~ be an arbitrary complex 2orel measure on [a,b] and let @ ~ L i,q. Let F be defined as in (2.3). Set u(t) = K~(F)* for t ~ [a,b]. Then, for all t ~ [a,b], u = u(t) satisfies the following Volterra-Stieltjes integral equation: -(t-a)(HO/~) u(t) = e -(t-s)(H0/k) + /[a,t) e @ ,n(s) u(s) n(ds) , (4.25) where e is given by (2.5) . The Bochner integral Yn (4,25) converges in the strong operator topology; that is, for all r ~ L2(R N) and t ~ [a,b],

-(t-a)(H0/l) u(t)~ = e -(t-s)(Ho/k) + f[a,t) e @ ,n(s) u(s)$ ~(ds) . (4.26)

PROOF. Fix k ~ ~ and t ~ [a,b] . We use the approximation procedure presented in Section 4.B. We briefly recall the notation. Given a positive integer, let nm = p + Emp=l Up~p be as in (4.9) and let F m be as in (2.3) except with n replaced with Dm; let Um(t) = K~(Fm)* According to Theorem 3.2, we have:

-(t-a)(Ho/X) Um(t) - e 186 Lapidus -(t-s)(Ho/k) = I[a,t) e 8~,nm(s) Um(S)~m(dS) . (4.27)

In view of (4.25), we want to show that -(t-a)(H0/%) u(t) - e

-(t-s)(HOlX) = f[a,t) e B ,~(s)u(s)n(ds) . (4.28)

The fact that the right-hand side of (4,28) is actually well defined as an element of L(L2(~N)) will be justified in Lemma 4.1 below. Note that, by Theorem 4.2, Um(t) converges to u(t) in the norm operator topology. In order to establish (4.28), it thus suffices to show that the right-hand side of (4.277 converges to the right-hand side of (4.287 in the norm operator topology [Y.e., in L(L2(EN)]. We now write: -(t-s)(Ho/~) f[a,t) e 0,n(s)u(s)~(ds)

-(t-s)(Ho/k) f[a,t) e e~,nm(S) Um(S)~m(ds)

= Bl,m + B2,m + B 3 ,m ' (4.29) where -(t-s)(H0/%) Bl,m = f[a,t) e [8 ,D(s) - 8 ,nm(S)]U(S)n(ds) , (4.30a)

-(t-s)(Ho/k) B2,m = f[a,t) e 8 ,~m(S)[U(S) - Um(S)]N(ds) , (4.30b) and -(t-s)(Ho/k) B3,m = f[a,t) e 0~,nm(S) Um(S) (n - nm)(ds) (4.30c) Lapidus 187

Note that the integral defining Bl,m, B2, m or B3, m only converges in the strong operator topology [i.e., when applied to an arbitrary vector ,~ in L2(~N)]; the reader should keep this remark in mind when looking at equation (4.39), (4.41) and (4.45) below. We next show that Bl,m, B2, m and B3, m tend to zero in the norm operator topology as m 4 ~. In view of (4.29) and the discussion preceding it, this will imply (4.28) and thus establish Theorem 4.3. We first study BI, m According to (2.57. ,

0 ,n(s) - 0 ,nm(s) = [~(n(s)0(s)) - ~(nm(S)0(s))] e(s) .

Since by (4.1) and (4.9), l](s) = qm(S) for s r {~m+l' ~m+2' "''} ' and n([ ) = ~ , we have: p p

i if S ~ [a,b]\l~m+l,... } O,,n(s) - O,,nm(S) = ~(~pe(~p)) - I] o(~ ) if s = p p with p ~ m+l . (4.31) Thus, by (4.30a) and (4.31),

-(t-~p)(H0/~) = e [~(~pS(~p)) - I] Up 0(~p)U(~p). (4.32) Bl'm p=m+l C[a,t) P Note that the summation is taken over those 's such that p > m+l and P { last) . P Now, by (2.5) and (2.9),

~(~pS(~p)) ~p0(~p) = mpe ,D(~ p) = e ~ P ~(~p) -I; hence (4.32) becomes

E e-(t-~p)(Ho/l) [e ~pS(~p) Bl,m - I - ~pS(~p)] U(~p) . (4.33) p=m+l ~p~[a,t)

It is elementary to check that

e X - 1 - x <_ xe x , for all x > O; 188 Lapidus since L(L2(EN)) is a Banach algebra, it follows that for all p ~ i, ][empO(~p) Imp[I[8( - I - ~pe(~p)[I ~ [~pl~e(~p)~ e ~P)[[ (4.34)

Clearly, Impllle(~p)~ < liell~l;n < +~ (4.35) since 0 ~ Lml;n and b lleH~l;n = af le(s)llipl(ds) + q=lZ I~qlUe(~q)ll. (4.36)

Thus (4.34) yields, for all p t I,

Ue~pe(~p ) 11ell~l;n - I - mpe(~p)H ~ ;mpIlle(~p)U e (4.37)

II e I1~1; n Since by (4.8), llu(Tp)I[ ~ e for p ~ i, it follows from (4.33) and (4.37) that 2[lellml;n HBI,mlI ~ ( ~ lmpllie(~p)[[) e (4.38) p=m+l

Since % ~ Lml;D , the series Z~=iI~plUe(~p)II is convergent and thus (4.38) implies that BI, m tends to 0 in L(L2(EN)) as m~m Next we consider B2, m By (4.305) ,

t -(t-s)(HOIX) B2'm = a/ e O~,nm(S)[U(S) - Um(S)]p(ds),

m -(t-~p)(H0/k ) + Z e ~p8 nm(~ p) [U(~p) - Um(~p)]. (4.3~) p=l ~p~[a,t) By (2.8) and (2.9),

e (s) = e(s) for s ~ [a,b]\{~ I ..... ~m} (4.40a) ~,n m and = e~p0(~p) - I for i ! p ~ m . (4.40b) ~p@~,qm(~p ) Lapidus 189

Thus (4.39) becomes:

t -(t-s)(H0/k) B2,m = I e O(s)[u(s) - Um(S)]p(ds), a -(t-%p)(Ho/k) E e mpO(%p) [U(%p) - Um(%p)] p=m+l ~p~[a,t)

m -(t-~p)(H01k) s e [e ~pS(~p) - I] [U(~p) - Um(~p)]. (4.41) p:l ~p~[a,t)

According to (4.10),

sup Ilu(s) - Um(S)ll ~ c m (4.42) s~[a,t)

Moreover, one can see easily that

e X - 1 -< xe X , for all x > 0 ,

thus, in view of (4.35), we have for all p 2 1 , lle~pO(~p) [[le(~p)ll - Ill ~ I~pllle(=p)ll e Imp Iloll.1;n I%llle(~p)ll e (4.43)

It now follows from (4.41)-(4.43) that t llB2'mli ~ (fa llS(s)lllPI(ds)) cm + (p:m+l~implliS(~p)ll) Cm

m Ilell~l;n + ( • I~apllle(~p)ll) e c p:l m

b (I Ile(s)lll~l(ds) + C s l~plile(~p)ll) e m a p:l 190 Lapidus

In view of (4.36), this yields II e I1| n e C (4.44) IIB2,mll <-. IIOIl(=l; n m

Since, by Theorem 4.2, c q 0 in m as m ~ ~ , (4.44) implies that B2, m ~ 0 L(L2(~N)). Finally, we examine B3, m . By (4.1) and (4.9) ,

= q - qm p=m+l~ ~P ~ ~p

Hence, by (4.30c) and (4.40b) ,

-(t-~p)(H0/~ ) [e~pB(~p) (4.45) B3'm = p=m+lZ e - I]Um(~ p) . ~p~[a,t)

It now follows from (4.8), (4.43) and (4.45) that 2 II e I1,~1; n liB3,m] I <_. ( T, I~pttle('~p)U) e (4.46) p=m+l

Since 8 ~ L l;q , (4.46) implies that B3, m -~0 in L(L2(~N)) as m ~ ~ .

In light of (4.28) and (4.29), this concludes the proof of Theorem 4.3. m

In the statement and the proof of Theorem 4.3, we have used the following result.

I~ 4.1, Let k ~ ~; Let q be a Borel measure on [a,b] and let O fi Lml;q Then 8@,q fi L l;q and Eor every t ~ [a,b],

-(t-s)(H0/k) f[a,t) e e@,q(s) u(s) q(ds) fs a strong Bochner integral and hence defines an element of L(L2(~N)) .

PROOF. It follows from Lemma 2.1(c) that 0 belongs to r L~l;q and that II e Ilcol; n II%,nll~l;n _< e -1 . (4.47) Lapidus 191

Fix , $ L2(| N) , t 6 [a,b] and I 6 ~ . Set -(t-s)(H0/l) f(s) = e O,n(s) u(s)# , for s ~ [a,t) . (4.48)

We want to show that f is Bochner integrable with respect to q . According to Bochner's integrability criterion [ii, Theorem 3.7.4, p. 80], it suffices to show that f is Bochner measurable on [a,t) and that ~IfN is Lebesgue integrable with respect to IDI on [a,t) . Let u be defined as in Section 4.B. Recall that by [18, m Theorems 2.1 and 2.3, p. 98], u m is strongly Bochner integrable on [a,b]; in view of Theorem 4.2, the same is true for u . (Actually, it can be shown in a llke manner that u has left and right strong limits at every point of [a,b]; see Theorem 5.2 and Remark 5.5(a) below.) Since the contraction semigroup {e -o(HO/I) }p>O is strongly continuous, it follows easily that f is Bochner measurable on [a,t). Now, by (4.47) and (4.8), llf(s)n ~ 8e~,n(s)ll~u(s)~n~ lle~,n(s)ll ellell| II~II (4.49)

According to (4.47)-(4.49) ,

/[a,t) llf(s)lllnl(ds) ~ Ne~,nll~l;n ellea| II~II 211eli| lleH| ) (e ;n - e ,~II < +~ , (4.50) as desired. D

COROLLARY 4.1. Under the assumptions of Theorem 4.3, we have for all t ~ [a,b]:

-(t-a)(BO/t) 211ell| lle~| llu(t) - e II ~ e - e (4.51)

PROOF. This follows immediately from (4.26), (4.48) and (4.50). m 192 Lapidus

Remark 4.3. The integral equation obtained in Theorem 4.3 is reminiscent of a Duhamel type formula or "sum rule", in the sense of [4], for product integrals of measures. In order to make this statement more specific, we need to extend the results of [4, Chapter V] to the present case of strong product integrals of measures in the context of infinite dimensional spaces. (In a special case, this was announced in [17, Theorem 1.4, p. 6].) We refer the interested reader to [20] where this subject is further developed.

5. STUDYOF THE INTEGRAL EQUATION AND OF ITS SOLUTION

5.A. Uniqueness of the Solution

We show in this section that the integral equation obtained above has a unique bounded solution, necessarily equal to u(t) .

TIIEOREM 5.1. Let )t ~ ~+ . Under the assumptions of Theorem 4.3, the VoTterra-StYeTtjes integral equation -(t-a)(H01~) v(t) = e -(t-s)(H0/k) + l[a,t) e 0r v(s) ~(ds), (5.1) for t fi [a,b] , has a unique bounded so]ution v: [a,b] ~ L(L2(RN)) . This soTution is given by v(t) = u(t) = K~(F)* , for t [a,b] where u is defined as fn Theorem 4.3, Moreover, for all t G [a,b], u(t) is given by the following time-ordered eATonentYa] series '9

u(t) = E f Ln,n(s I ..... Sn;t) (~x...x~)(ds I .... ,dSn), (5.2) n=O AS(t) where

A~(t) = {(s I .... ,s n) 6 [a,t)n: a 5 s I 5...5 sn < t} (5.3) and for (Sl,...,Sn) ~ A'n(t) , Lapidus 193

L n,n = L n,q (s i"'" Sn;t)

-(t-Sn)(Ho/~) -(Sn-Sn_t)(Ho/k) = e O,ii(s n) e O,q(Sn_l)...

-(s2-sl)(H0/X) -(sl-a)(HO/X) O ,q(s 2) e O ,n(sI) e (5.4)

The serYes fn (5,2) fs absofutelT~ummabJe Yn L(L2(~N)), uniformlv for t g [a,bj; ~trther, the Jntegz~7 in (5.2) is a strong 2ochner integraJ,

PROOF. Let v be a solution of the integral equation (5.1) such that l[v(t)I[ ! M for all t in [a,b] . After n-I iterations of (5.]), we obtain n v(t) - l fk~(t) L~,n n(dSl)'"n(ds~) ~=0

-(t-Sn+l)(HO/%) -(Sn+l-Sn)(Ho/k) 8 ~( = fA~+l(t) e 8,n(Sn+ I) e ~, Sn)

.. O~,~(s t) v(s 1) ~(dSl)...~(dSn+ I) , where 5' or L ,q is given as in (5,3) or (5.4), respectively. Consequently,

n IN(t) - I: fg'(t) I%,n (n•215 ..... d%)ll

M fS~+l(t) Ilen(Sn+l)ll...lfen(sl)ll (Inl•215 I .... ,asn+l)

1 < M ~ (/[a,t) IIO

1 )n+l < M 1 II~ 1)n+l M~(IIo nil<; n _ ~(e - , (5.5) where we have used (2.11) in the last step of (5.5). Since 0 g L~I;~ it follows that 194 Lapidus

v(t) = fAd(t) Ln (nx...xn)(ds I ..... ds n) , in L(L2(~N)) . n=O ,q

This shows that (5.1) has a unique bounded solution on la,b] . Since 6y (4.8) and Theorem 4.3, u is a bounded solution of (5.1) on [a,b] , it must necessarily be the unique solution of (5.1). o

COROLLARY 5.1 Under the assuraptYons of Theorem 4, 3, we have for all nonne~ative integers n ,

n

sup Ilu(t) - S ]h,a(t) La,q(s 1 ..... Sa;t) (O•215 1 ..... dsa)l] U=O

lleIl| I (ller II )n+l < e ll~ i (elleIl~1;n i)~+I e (n+l)! q ml;n - (n+l)!

PROOF. This follows from the proof of Theorem 5.1; note that in view of (4.8), the second inequality is obtained by letting M = exp(USll~l;q ) in (5.5). o

Remarks 5.1. (a) It follows in particular from Theorem 5.1 that for each t ~ [a,b] , the time-ordered exponential series in (4.3) and (5.2) are equal. This was certainly not obvious a prforJ. Note that the "disentangling process", in the sense of Feynman's operational calculus [7], is carried out much further in the generalized Dyson series in (4.3) than in the series in (5.2). In the case of a measure with finitely supported discrete part studied in Section 3, a similar remark applies to the generalized Dyson series in [13, Eq. (3.5), p. 36] or [18, Eq. (4.1), p. II0], on the one hand, and the series in (5.2), on the other hand. (b) The series in (5.2) is nothing hut the Neumann-Liouville series (or iterative solution) of the Volterra-Stieltjes integral equation (5.1). (Compare with the series in [4, Theorem 3.2, p. 164].) Lapidus 195

The next result shows that the series in (4.37 and (5.27, although they have the same sum according to Theorem 5.1, are usually not equal term hy term.

LEMMA 5.1. For ~ ~ ~ and t ~ [a,b] , let U(n)(t ) := K~(F(n))* , ~here for a nonnegatYve ~nteger n , the functlona] F(n ) fs deffnedby

F(n)(X) = n~ ([[a,t) 0(s'x(s))n(ds))n ' x ~ C. (5.67

Then U(n)(t7 Js equal to the n-th term oft he series Yn (4.3); however, Yt fs not Yn ~eneral equal to the n-th term of the serYes Yn (5.2).

PROOF. First we note that by construction [13], U(n)(t) is equal to the n-th term of the series in (4.37. Next we give a simple example illustrating the second statement. Let q = B + ~ with m # 0 and ~ < t. Then -(t-s)(Ho/X) -(s-a)(HoIX) u(1)(t) = f[a,t)e 0(s)e n(ds)

-(t-s)(H0/[) -(s-a)(Ho/[) = f[a,t)e e(s)e B(ds)

-(t-~)(H0/~) -(~-a)(Ho/~) + e [~8(t)]e

whereas, by (2.97, the first term of the series in (5.2) is equal to -(t-s)(Hol[) -(s-a)(Ho/%) f[a,t)e 0 q(s)e q(ds)

-(t-~)(Ho/k) eOe(t) -(z-a)(Ho/k) + e [ - I] e

Remarks 5.2. (a) I wish to thank Professor Gerald W. Johnson for providing me with the above counterexample after having read the preprint of this paper. 196 Lapidus

(b) Naturally, if q = p is a continuous measure, the series in (4.3) and (5.2) are equal term by term. (See Eq. (2.8) and [18, Eq. 3.7), p. 102].)

5.B. SpecYal Cases off the Integral EquatYon

We discuss here various special cases of the integral equation (4.25) [or equivalently, (5.1), with v = u]. In the following, q = p + is the unique decomposition of ~ ~ M([a,b]) into its continuous part p and its discrete part ~

5.B.I. Purely Continuous Measure: ~ = p

Assume that n = p is a purely continuous measure (f.e., ~ = 0) . Since q does not have any pure point in this case, Lemma 2.1(a) implies that

e ,q(s) = Br = s , q-a.e, in [a,b) . (5.7)

Hence the integral equation (4.25) becomes for all t ~ [a,b] : -(t-a)(H0/k) t -(t-a)(H0/k) u(t) = e + fa e B(s) u(s) p(ds) ; (5.8) this was already obtained in [17, Eq, (1.12), p. 6] or [18, Corollary 2.1, pp. 98-99]. In particular, if n = # = ~ , where ~ denotes ordinary Lebesgue measure on [a,b] , we recover the usual integral equation associated with either the heat equation in the diffusion case (k = 1 and 8 = -V) -(t-a)Ho t e-(t-a)H0 u(t) = e - fa V(s) u(s)ds (5.9a) or the Schr8dinger equation in the quantum mechanical case (k = -i and

8 = -iV) -i(t-a)Ho _r -i(t-a)Ho u(t) = e - i J~ e V(s) u(s)ds . (5.9b)

Recall that in light of (2.2) and (2.3), (5.9a) yields the classical Feynman-Kac formula while (5.9b) gives one interpretation of the Feynman path integral. We refer to [18, w pp. 124-125] for a further discussion of this case. (In (5.9), it is customary to let a = 0 when the "potential" V is time independent.) Lapidus 197

If the measure p is absolutely continuous, (5.8) becomes

-(t-a)(Boll) t -(t-a)Holl) u(t) = e + [a e ~(s) u(s)ds , (5.10) where 8(s) := (dB/ds)(s) 8(s) and (dB/ds)(s) denotes the density of (that is, the Radon-Nikodym derivative of B with respect to ~) . In particular, in the diffusion or quantum mechanical case, we obtain the same formula as in (5.9a) or (5.9b), respectively, except with V(s,.) replaced by the new potential V(s,.) := (d~/ds)(s) V(s,.) . (See [18, Corollary 5.1, p. 180] and [13, pp. 23-24] for a physical interpretation of this case.) In general, however, n = B has both an absolutely continuous part and a (continuous) singular part (see, e.g., [I, p. 141; 24, pp. 20-27; 27, pp. 35, 116 and 180]. It would be interesting to find a "realistic" physical interpretation for the case of a purely singular ~ , for instance. Possible candidates include Schr~dinger operators with almost periodic potentials or certain distributions that occur in statistics or probability theory. If P is the Lebesgue-Stieltjes measure associated with a Cantor-like function, say, approximation by absolutely continuous measures with step density functions, in the spirit of [13, w should help to develop further intuition in this setting. The use of singular measures - such as Cantor-Lebesgue measures - may also be relevant to study aspects of quantum chaos (e.g., the chaotic quantum oscillator or pendulum), as well as to introduce the notion of "fractal time" (e.g., [23, Chapter 6]) in the context of path integrals. (This comment suggests some interesting connections with recent work of the author [21] where aspects of "fractality" in space are examined.) Note that for a singular continuous measure, d~/ds = 0 ~-a.e. and the situation is apparently quite different from that of an absolutely continuous measure; however, both situations are described by the same integral equation (5.8).

Remark 5.3. In light of (5.7), the perturbation expansion in (5.2) reduces in this case to the "usual" generalized Dyson series associated with a continuous measure. (See [13, Corollary 1.2, pp. 22-23, and Remark 1.5, p. 25] and [18, Eq. (3.6), p. 102].) 198 Lapidus

5.B.II. Measure with Finitely Supported Discrete Part

Assume that n = p + ~ , with ~ continuous and ~ = Em p=l Up %P with ~0 := a N ~I <'"< ~m b =: ~m+l " This is the case studied in [18] and re-examined in Section 3. By virtue of the uniqueness of the solution (Theorems 3.2 and 5.1), the integral equation (5.1) [with v replaced by u] becomes in this case -(t-~p)(H0/X) UpB(~p) u(t) = e e U(~p) -(t-s)(Ho/~) + ]$ e 8(s) u(s) p(ds) , P for all t ~ (~p,~p+l ] and each p = O,...,m . (5.11)

Actually, it is worthwhile to give a direct and more formal derivation of this fact. This will show in particular that (5.11) and (5.1) are truly equivalent equations in this situation.

PROPOSITION 5.1. Suppose that n fs gYven as above. Let ~; and let v: [a,b] -) L(L2(RN)) . If v Ys a solutYon of (5.1), then it is a solutYon of (5.11). Conversely, every solution of (5.11) satisfies (5.1).

PROOF. Assume that v is a solution of (5.1). To emphaNize the -P(H0/k) formal aspects, we set T(p) = e for p ~ 0 ; note that {T(p)}p~0 is a semigroup. Given p ~ {O,...,m} , let t ~ (~p,~p+l ] . Then we write successively: v(t) = T(t-a) + f[a,t) T(t-s) 0 ,n(s) v(s) n(ds) = T(t-a) + f[a,~p) T(t-s) 8 ,n(s) v(s) ~(ds) u e(~ ) + T(t-~ ) (e p P - I) v(~_) t P P + f~ T(t-s) O(s) v(s) ~(ds) ; (5.12) P in the second equality of (5.12), we have used Lemma 2.1 according to which mpO(Tp) Up O,n(T p) = e - I and O,N(s) = O(s) for s ~ (~p,t) . Lapidus 199

Now, by (5.1) with t = ~ , P

V(~p) = T(~p-a) + f[a,Xp) r(~p-S) 0r v(s) n(ds)

and hence, by the semigroup property of T(.) and [II, Theorem 3.7.12, p.83],

T(t-Xp) V(~p) = T(t-a) + f[a,!p) T(t-s)e ,~(s) v(s) n(ds) (5.13)

By combining (5.12) and (5.13), we obtain:

v(t) = T(t-~p) e ~p0(~p) V(~p) + f~ T(t-s) O(s) v(s) p(ds) ; P this shows that v satisfies (5.11). The converse follows from the proof of Theorem 3.2 if we note that the fact that u was a bounded solution of (5.11) was not used explicitly in that proof. D

Remarks 5.4. (a) The proof of Proposition 5.1 suggests possible extensions of parts of the present framework to more general Banach algebras than L(L2(I N) , as well as to yon Neumann algebras, for instance. We expect to pursue this topic in a later work. (See also [20, 22].) (b) The proof of Proposition 5.1 also shows that the integral equation in [4, Theorem 3.1, p. 161] can be made much more explicit in the case of a measure with finitely supported discrete part. (c) It would be interesting to compare aspects of the present work and [13, 17-18, 20] with the recent approach to path integrals involving the notion of Poisson processes (see, e.g., [26] and references therein). Although the two theories have different scope and have been developed quite independently as well as with distinct purposes in mind, we feel that such a comparison - which is not attempted here - would be mutually beneficial.

In [18, Theorem 2.4, p. 99 and Section 6.B, pp. 125-127], we give an explicit expression for u(t) in the present situation of a measure with finitely supported discrete part; namely, for p ~ {O,...,m} and t ~(~p,~p+l ] , 200 Lapidus

~pe(~p) ~p_l~(~p_ I ) = e P(~p,lp_l)e u(t) P(t,~p) a10(~1) P(~l,a) , (5.14) .,. P(~2,~I) e where for b 2 t 2 2 t I 2 a , P(t2,t I) denotes the "propagator" for the integral equation (5.8) associated with the continuous part p of ~ ; (that is, P(t2,t I) sends the solution of (5.8) at tzme t I onto its solution at time t2.) Possible physical interpretations of this result both in the quantum mechanical and the diffusion cases are provided in [18, Sections 6.C.I and 6.C.II, pp. 127-129]. We simply recall here that in the quantum mechanical case and for B = ~ and t C (lm,b] , we may interpret (5.14) as corresponding, in particular, to the scattering of a nonrelativistic particle at successives times ~1,...,~m .

5.B.III. Pure!y Discrete Measure: q =

Assume first that ~ = v is a discrete measure (f.e., ~ = 0) with finite support: v = ~mp=l ~p~ with t 0 := a ...... < ~I < < ~m -< b =: ~m+l P According to (5.14), we then have for p ~ {0,...,m} and t ~ (~p,~p+l]:

-(t-~p)(H0/k)e~pe(~ p) e-(~p-~p_l)(Ho/I) ~p_10(~p_ I) u(t) = e e ~le(~l ) -(~l-a)(Ho/X) ... e e (5.15)

The following special case is of particular interest. Assume that a = 0 , b = t , ~ = p(t/~) and ~ = = t/m for p = I~ ..,m " so that ~ = m p p ~p-~p-i " ' (t/m) ~p=l ~p(t/m) ' {Note that b = Xm = t ~ (~m_l,~m] and that, as mentioned in Remark 2.1(d), we use here the fact that n may have mass at b.) Then, for time independent potential, (5.15) yields

Kt.(F) = u(t)* = (e -i(t/m)H0 e -i(t/m)V)m (5.16) -i or

(5.17 K~(F) = u(t)* = (e -(tlm)HO e -(tlm)V)m , Lapidus 201 in the quantum mechanical or diffusion case,respectively. (See [18, Corollary 2.2, p. 100].) We recognize in (5.16) or (5.17) the familiar Trotter products for Schr~dinger unitary groups or for heat semigroups, respectively. (For further connections with the Trotter product formula, see [13, Example 4.1, pp. 54-56].) Assume now that ~ = ~ is an arbitrary discrete measure: = ~p=l mp6 with {~p} C [a,b] and Ep=l~ [~pl < +~ Then, in view P of (2.9), the integral equation (4.25) becomes:

-(t-a)(Ho/k ) ~ -(t-~p)(H0/l) (e~pe(~p) u(t) = e + ~ e - l)U(~p) . (5.18) p=l ~p~[a,t)

In general, the sequence {~p}p=l cannot be ordered or can accumulate at some point(s) of [a,b] . (For instance, {~p} could be an enumeration of all the rationals in [a,b]; so that the support of ~ is all of [a,b] . In the spirit of [18, w pp. 127-128] and in view of Theorem 5.2 below, we could then interpret u(t) in the quantum mechanical case as corresponding to a particle scattered at every rational time and freely propagating "in between".) Thanks to our basic approximation theorem, it is possible to write u(t) as a limit of expressions of the type (5.15). In order to avoid using cumbersome notation, we consider instead the following simpler situation. Suppose further that {~p}~=l_ is a strictly increasing sequence, with limit denoted by T . Given t ~ (a,b] , we distinguish two cases: First assume that t < T . Then, there exists a unique p ~ 0 such that t ~ (~p,~p+l ] . (We use here the convention t 0 = a.) We claim that u(t) is still given by (5.15) in this case. In fact, the restriction of v to [a,t) is equal to EPq=l ~q~ and hence the result recalled at the q beginning of this section applies. Next assume that t > T . Then we claim that u(t) is equal to the norm limit of Up(t) as p ~ ~ , where Up(t) is given by the right-hand side of (5.15). Indeed, for every fixed p ~ 0 , we have t > ~ ; so that P Up(t) = K~(Fp)* , as defined in Section 4.B, is equal to the right-hand side of (5.15). The claim now follows from Theorem 4.2. 202 Lapidus

5.B.IV. General Borel measure

Let ~ = P + ~ be an arbitrary Borel measure on [a,b] . In order to avoid repetitions, we may assume that ~ = Emp=l ~p6 is not finitely P supported. Since the situation exhibits both "continuous" and "discrete" features, much of the discussion of Sections 5.B.I and 5.B.III can be adapted to this case. For example, for t ~ [a,b] , the integral equation (4.25) can be rewritten in the form:

-(t-a)(H01%) t e-(t-s)(Ho/X) u(t) = e + fa 8(s)u(s)p(ds)

-(t-~p)(H0/X) (empS(~p) + Z e - I) U(~p) . (5.19) p=l ~p~[a,t)

Suppose in addition that {~p}p=l is strictly increasing, with limit denoted T . From a physical point of view, this is a very natural assumption. Then, by means of (5.14) and Theorem 4.2, one sees just as at the end of Section 5.B.III that for a given t ~ (a,b] , we have the following two cases: either t < T and then u(t) is given by (5.14), where p is the unique nonnegative integer such that t ~ (~p,~p+l ] ; or t ~ T and then u(t) is the norm limit of Up(t) as p ~ ~ , where Up(t) is equal to the right-hand side of (5.14). In the spirit of [18, Section 6.C.I] and when p = ~ , we may interpret the latter result in the quantum mechanical case as a countably infinite number of scatterings (or "shocks", or "instantaneous interactions") occurring at the successive times ~I,~2, .... Additional properties of u(t) are given in Sections 5.C and 6 below.

Remark 5.5. We mention without going into the details here that a dYstributional differential equation in some sense equivalent to the integral equation (4.25) can be obtained for a general measure ~ . (See [20, esp. Theorem 2.5] for the quantum mechanical case.) Lapidus 203

5.C. Concrete Form of the Zntexra] Equation

We give here a more concrete expression for the integral equation (4.25) of Theorem 4.3; this form may look more familiar to some readers. First recall the well known integral representation of the free semigroup (see, e.g., [13, Eq. (0,7), p. 7]) . For ~ ~ L2(EN) , ~IN,~r ands>0 ,

-s(H0/X) [ X lN/2 (e *)(~) = 12-~st flN e 2s #(q) dNq , (5.20)

% J where dNq denotes Lebesgue measure in E N and the integral in (5.20) is a Lebesgue integral for I ~ ~+ but must he intepreted in the mean, as in the theory of the Fourier transform, for ~ not integrable and purely imaginary. Now, given @ ~ L2(EN) , we set ~t = u(t)~ . Then (4.26) yields for all t ~ [a,b] and (almost every) ~ ~ E N :

~t(~) __ [~} f|N e 2(t-a)#(q) dNq

/ X )N/2 -~lq-El 2 2(t-s) + /[a,t) f,N [~] e 0 ,n(S,q ) ~S(q) dNq n(ds) , (5.21) where the function ~ ,~(-,-) is given by (2.7). We leave it to the reader to rewrite (5.21) in the diffusion and quantum mechanical cases.

5.D. DYscontJnuJties oft he Solution

The following result shows that u, as a function of time, is in general discontinuous at a pure point of n . 204 Lapidus

I~IEOI~ 5.2. Fix k fi ~; . s q be an arhYtrary Bore] measure in M([a,b]) and Jet u(t) = K~(F)* be the unique solution of (5.1) as in Theorem 5.2. Then u is strongly ]eft-contYnuous on (a,b] and is strongly continuous at every point of [a,b] ~h~ch is not a pure point of q . Horeover, for every g [a,b) such that ~ := n(~) # 0 , we have:

u(~+) = e ~(~) u(~) , (5.22)

~here u(~+) denotes the strong 11mlt of u(t) as t e ~ , t > % (Y.e., for every ~ G L2(R N) , limt_h,t> ~ u(t)~ = e mB(~) u(~)~) .

PROOF. Let q = B + ~ and n = P + vm be as in (4.17 and (4.97, respectively; recall that ~ = Ep=l m ~p~p and vm = I mp=l ~p6 Set P Um(t) = K~(Fm)* as in Section 4.B. According to Theorem 4.2, Um(.) converges to u(-) , unYformTyin [a,b] , in the norm and a fortYorf in the strong operator topology. The first part of the theorem follows since, by [18, Theorem 2.3, p. 98], u is strongly left-continuous on (a~b] and strongly m continuous at every point which is not a pure point of ~m and hence of B Now, let ~ he a pure point of q with m := q(1) ; we may assume that ~ = ~mo for some m 0 ~ I . According to [18, Theorem 2.31, we have

Um(~+ ) = e ~8(~) Um(~) , for all m _> m 0 ; note that ~ = qm(~) = q(~) for m > m O It follows that u(~+) exists and that

u(~+) = lim Um(~+) = e m8(1) u(~) , m-~ as required. Once again, the interchange of limits is justified by the uniform convergence of u m to u . []

Remarks 5.6. (a) The advantage of the ahove proof is that it does not rely on the integral equation of Theorem 4.3. This fact was used in the proof of Lemma 4.1. We now give an alternate proof. By Theorem 4.3, -(t-a)(Ho/k) -(t-s)(Ho/k) u(t) - e = f[a,t) e O q(s) u(s) q(ds) Lapidus 205

-t(Ho/%) By letting t ~ x , t > and since e is strongly continuous, it follows that

u(x+) - u(x) = o,,n(x) u(x) n(~) = (e n(x)~ - I) u(x) .

Hence the result. (b) As a special case of Theorem 4.2, we see that if n = P is a continuous measure, then u is strongly continuous on all of [a,b] . This is why in the usual situation where n = ~ is Lebesgue measure, this phenomenon of discontinuity had not previously appeared in the literature.

We conclude this section by stating a related result.

THEOR~I 5.3. fix X ~ ~; . The function v dofinedby v(s) = -(s-a)(HO/X) u(s) - e fs strongly of bounded variation on [a,b] .

and let s O := s I <...< Sq_ 1 < b =: s PROOF. Let # ~ L2(R N) a < q be a partition of [a,b] . Then, according to (4.26) ,

q E llV(Sk) ~ - V(Sk_l)~l I k=l

q -(t-s)(Ho/k) O k=lZ II/[sk- I 'Sk) e ~'D(s) u(s)~ ~(ds)ll

lloll~l;" [ -< II*ll e k=l f[Sk-l'Sk ) [10,,n( s)lllnl(ds) t

IIo I1~1 ; n <_ II,llllo,,nll~l;n e < + ~ , here we have used (4.8) in the first inequality. This establishes Theorem 5.3. 206 Lapidus

6. FURTHER PROPERTIES OF THE PROPAGATOR

In this section, ~ will denote a complex-valued set function on (the Borel subsets of) [a,+m) whose restriction to [a,b] for any b > a is a complex Borel measure on [a,b]. Let e: [a,+~) • RN ~ ~ he a Borel measurable function whose restriction to [a,b] • RN for any b > a belongs to L~l;~([a,b)). Given tl,t 2 ~ [a,+m) with t 2 Z t I , let the "propagator" U(t2,t I) be defined by t 2 U(t2,tl) = Kk (F)* , for k ~ ~ , (6.1) where CO = C0([tl,t2)) and

F(x) := exp(f[tl,t2) 8(s,x(s)) n(ds)) . (6.2)

[Remark that with the previous notation, u(t) = U(t,a) for t ~ a.] According to Theorem 5.1, U(.,t I) is the unique hounded solution of the integral equation (5.1), where [a,b] is replaced by [tl,t2] ; in particular, -(t2-tl)(H0/k) U(t2,t I) = e -(t2-s)(H0/k) + f[tl,t2) e 0 ,~(s) U(s,t I) ~(ds) . (6.3)

The next result shows that U(.,.) is the propagator for the above integral equation .

TBEOREM 6.1. Assume that t 3 ~ t 2 ~ t I ~ a . Then U(tl,t I) = I and

U(t3,t I) = U(t3,t 2) U(t2,t I) (6.4)

PROOF. Eq. (6.4) holds for a measure with finitely supported discrete part, as was seen in [18, w In view of Theorem 4.2, the result then follows by passage to the limit, m

Remark 6.1. Theorem 6.1 can also be deduced directly from [14, Theorem 2 (or Theorem 4)] if we note the exponential property Lapidus 207

exp(f[tl ,t3 ) O(s,x(s)) n(ds))

= exp(f[tl,t2) 0(s,x(s)) n(ds)) exp(f[t2,t3) O(s,x(s)) ~(ds))

We note that [14, Theorem 4] was motivated by some of the results in [18], especially Theorem 2.4, p. 99.

An interesting special case is considered in the next theorem.

TBEOREM 6.2. Let tl,t 2 vYth t 2 2 t I 2 a . Zet p + v be the decomposition of the restriction of ~ to [tl,t 2) into its continuous and discrete parts. Assume thaC p = ~ , the ordinary Lebesgue measure on [tl,t 2) , that l~ is real and V in L;l,q([tl,t2)) is real-valued. Then, fn the quantum mechanical ease (k = -i and 8 = -iV) , U(t2,t I) is a unitaryoperator.

PROOF. According to [18, Proposition 6.1, p. 126], this is true when ~ is finitely supported. The result thus follows from Theorem 4.2 by approximating ~ by finitely supported measures as in (4.9) since a norm limit of unitary operators is unitary, o

Remark 6.2. In the diffusion case, when p = ~ and for time independent real-valued potential, it is not true in general that U(t2,tl) is self-adjoint; this is due to the fact that n , unlike ordinary Lebesgue measure, is not necessarily invariant under time reversal. (See [18, Remark 3.2, p. 103].)

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Department o~ Mathematics The UniversYty of Georgia Boyd Graduate StudYes Research Center Atheus, Georgia 30602, USA

Submitted: May 15, 1988 Integral Equations 0378-620X/89/020211-1651.50+0.20/0 and Operator Theory (c) 1989 Birkh~user Verlag, Basel Vol. 12 (1989)

ON SOME CLASSES OF UNBOUNDED OPERATORS

Sch~ichi Ota and Konrad Schmudgen

Hyponormality, normality and subnormality for unbounded operators on Hilbert space are investigated and quasi- similarity of such operators is discussed.

1. INTRODUCTION

In this paper, we generalized some notions of bounded

operators to unbounded operators on Hilbert space, and we extend

some familiar results on quasi-similar bounded operators to un-

bounded operators. We will prove in Sect.3 that quasi-similar

closed formally hyponormal(which is an unbounded version of

hyponormality) operators have equal spectra. We will also show

in Sect.4 that the Douglas theorem ([6], Lemma 4.1) holds for un-

bounded normal operators; that is, quasi-similar normal(in

general, unbounded) operators are unitarily equivalent.

It seems to be of interest to know what kind of

properties on unbounded operators are preserved under quasi-

similarity. In particular, we lastly will prove that a closed

subnormal operator quasi-similar to a self-adjoint operator is self-adjoint. 212 Ota and Schmiidgen

2. QUASI-SIMILARITY

Let T be a linear transformation from a Hilbert space

into a Hilbert space X". Throughout this paper, the symbols

(T), ~(T) and Ker(T) stand for the domain of T, the range of T and the kernel of T, respectively. If T is a bounded

(everywhere-defined) linear transformation satisfying

Ker(T) = {0} and ~(T) = X ,

then T is simply said to be ......

Let A and B be operators in ~ and ~ respec-

tively, and let T be a bounded linear transformation from

into ~ . Then the relation

T-A c B.T means that T maps the domain ~(A) into the domain ~(B) and

T.A~ = B.T~ for all ~ ~ ~(A).

We now introduce the quasi-similarity of unbounded operators.

DEFINITION 2.1(Quasi-similarity). Let A and B be densely defined (in general, unbounded) linear operators in

Hilbert spaces ~ A and ~B, respectively. If there exist quasi-invertible (everywhere-defined) transformations XAB from

A into ~ B and XBA from ~ B into ~ A such that

XAB.A c B.XAB and XBA.B c A-XBA , then we say that A is q~a_s.i_:~.!m~[aT, to B and it is denoted by A ~ B.

LEMMA 2.2. The quasi-similarity is an equivalence ()ta and Schmtidgen 213 relation. Furthermore, if A ~ B, then A* ~ B*.

Proof. Suppose A ~ B. Since XAB is bounded, it follows that

A*.XAB* = (XAB.A)* D (B.XAB)* D XAB*.B* and analogously B*.XBA* D XBA*.A*. Hence A* ~ B*.

That quasi-slmilarity is an equivalence relation fol- lows in the same way as in the case of bounded operators.

REMARK 2.3. One can define the similarity of un- bounded operators in the same manner as in the case of bounded operators: If there exists a bounded linear transformation X from ~ A into ~B with bounded inverse such that

X.A=B.X

( which means that X maps ~(A) onto ~(B) and it satisfies

X.A~ = B.X~ for all ~ E ~(A)), then we say that A is s i$ilar to B with intertwining operator X and we write

A ~ B. If, in addition, the intertwining operator X is chosen as an isometry of ~ A onto ~B, then we say that A is u nitarily e~ivalent to B. Then, by the analogous argument to

Lemma 2.2, it is easily shown that the similarity is an equiv- alence relation, and that,if A ~ B, then A* ~ B*.

Thus some results in this paper may be stated in terms of similarity or unitarily equivalence. However we will be in-

terested in the behavior of quasi-similar operators, and so this reformulations will be left to the reader.

In the rest of this paper, an operator means a linear operator in a Hilbert space. 214 8ta and Scbmiidgen

PROPOSITION 2.4. An operator A quasi-similar to a closable densely defined operator B is also a closable densely defined operator, and moreover the closure A is quasi-similar to ~.

Proof. Let XAB and XBA be the quasi-invertible transformations such that

XAB.A c: B.XAB and XBA.B c A.XBA.

It is easily seen, by the relation XBA~(B) c ~(A), that A has a dense domain.

Suppose a sequence {x n} in ~(A) tends to 0 and a sequence {Ax n) tends to some 71 in ~ . Then one has

XABX n --> 0 and B.XABX n = XAB.AXn --~ XAB71 .

Since B is closable, it follows that XAB71 = 0, and so 71 = 0.

Hence A is closable. By Lemma 2.2, one has (A*) $ = A ~- (B$) ~

= B. This completes the proof.

Let A be a closed operator. A subspace ~ of ~ CA) is called a core for A if the closure of the restriction of A to is equal to A.

THEOREM 2.5. Let A and B be closed densely defined operators with bounded inverse. Suppose there exists a bounded linear transformation with dense range X such that

X'A C B.X.

Then X maps a core for A to a core for B, namely if ~ is a core for A, then X~ is also a core for B.

Proof. we first recall that, for a closed operator T, the domain ~ (T) with the inner product (~ta and Schmiidgen 215

( $ ,y )T = ( ~ , ~ ) + ( T~ , TN ) (~ ,~ ~ ~(T)) becomes a Hilbert space, denoted by (~(T), (.,-~).

Let ~ be a core for A. Since X is continuous as a transformation from a Hilbert space (~(A), (',')A } into a

Hilbert space (~(B), (.,.~ }, X~ is dense in X~(A) with respect to the graph topology of B. Hence we have only to show that X~(A) is a core for B, that is; it is dense in ~(B) with respect to the graph topology of B.

Let

T = B~ X~(A) be the restriction of B to X~(A). It suffices to show that

G(T) is dense in G(B) of ~ ~ ~ . Take (~ , B$ )E G(B) and suppose

( ($ , B$ }, (x, Tx) ) = 0 for all xE ~(T). It then follows that, for each 7/ E ~(A),

0 = ( $ , XT/ ) + ( B~ , B'XT/ )

= ( X*~ , A-I'AT] ) + ( B~ , X'A~ ).

Since ~(A) = ~ , it follows that

(A-1)*.X*~ + X*.B$ = O,

Since A -1 and B -1 are bounded(everywhere-defined) operators, it follows from our assumption that

X.A-1 = B-I.x.

Hence one has

X*.((B*) -1 + B )$ = O,

Since X has dense range, it follows that X* is injective, so that (B-1)*-B-1-B~ + B~ = O. It is clear that (B-1)*.B -1 + I is strictly positive. Hence B~ = O, and so ~ = O. This com- pletes the proof 216 6ta and Schmiidgen

REMARK 2.6. By a simple modification of the above proof, we see that the hypotheses on operators A and B that

each of them has bounded inverse can be replaced by the weaker condition : A is injective and has dense range, and ~(B)

c ~ (B *).

We now introduce for unbounded operators some analogous

notions of classes of bounded operators, which are useful in the sequel.

DEFINITION 2.7. A densely defined operator T is said

to be formally hyponormal if

(T) c ~(T*), and

(~ ~ ~(T));

and a formally hyponormal operator T is said to be b~YR~n__ozm~!

if

(T) = Z(T *).

Moreover, a formally hyponormal operator T is said to be for= mally normal if

11 T~ II = 11 T*~ II (~ : ~(T)); and a formally normal operator is said to be n~m__a~ if

~(T) = ~(T*).

Then, the following statements hold :

1. Let T be a densely defined linear operator. If T

is formally hyponormal (resp. formally normal), T is closable

and { is formally hyponormal (resp. formally normal). (~ta and Schmiidgen 217

2. For a closable densely defined operator T, T is

normal if and only if T and T* are both formally hyponormal.

3o A normal operator is closed and its adjoint is also

normal.

4. The inclusion relation of the classes cited above is as

follows;

Formally normal Self-adjoint c Normal c. C" Formally hyponormal ~'~Hyponormal

Clearly, all inclusions are proper, and the classes of formally

normal operators and hyponormal operators do not contain each

other.

Lastly, a densely defined linear operator T in a

Hilbert space ~ is said to be subnormal if there exist a

Hilbert space N containing ~ as a closed subspace and a

normal operator N in X such that

(N) ~ ~(T), and T~ = N~

for all ~ ~ ~(T) (see, for earlier works of such unbounded

operators, [1],t3],t12],E15],[16],[17] and the references cited

in them).

LEMMA 2.8. A subnormal operator is formally hyponormal.

~Too_f. Suppose T is a subnormal operator in a Hilbert

space ~ with a normal extension N in X , and let us take

in ~(T). For y ~ ~(T), one has

( T~ , ~ )~ = ( N~ , ~ )X = (~ , N*$ )X 218 Ota and Schmiidgen

Hence, @ in ~(T*). Moreover if P is the orthogonal projec-

tion of :~ onto ~ , then the above equalities implies

T*~ = P-N*~

for all ~ ~ (T). It follows that

II T~ II = II N~ II = II N*~ i[

II P.N*} II = II T*~ II ,

for all ~ ~ ~(T). Thus T is formally hyponormal.

REMARK 2.9. The class of subnormal operators has in general no relations with the class of formally normal operators with respect to the set-inclusion. This is easily checked by the examples in [31,[9] or [141. Of course, a subnormal operator in a Hitbert space which has a normal extension in the same Hilbert space is formally normal.

3. FORMALLY HYPONORMAL OPERATORS

LEMMA 3.1. If T is a formally hyponormal operator, then

II T~ II 2 ~ II T2~ II II ~ II for E ~ (T2).

Proof. For } E ~(T~), one has

I1 T~ II 2 = ( T*,T$ , ~ )

~_ II T*(T~ ) II [l ~ II

II T2~ II .ll $ II .

REMARK 3.2. If a densely defined operator T leaves 0ta and Schmiidgen 219

the domain invariant and it satisfies II T~ [I 2 ~ II T2~ II .[[ ~ [I

for ~ E ~(T), then T is said to be formally paranormal. A

formally paranormal operator is not always closable, and if it

is closable, then the closure is not necessarily formally

paranomal. In fact, it follows easily from the closed graph

theorem.

As is well-known in the case of bounded operators,

quasi-similarity and similarity do not preserve normality and

hyponormality. The following theorem is an unbounded version of

Clary's result(J2]) on bounded operators.

THEOREM 3.3. Let A be a closed formally hyponormal

operator in a Hilbert space ~ and let B be a closed densely

defined operator in a Hilbert space X . If there exists a

linear bounded transformation X with dense range from ~ to

such that X.B c A.X, then the spectrum of A is contained

in that of B.

Proof. Ne first note that, if A is formally hyponor-

mal and ~ E C (the field of all complex numbers), so is ~ .l

A. Hence, it suffices to show that if B is boundedly inver-

tible, so is A. Put ~ = II B-111 and take ~ E X .

Define

n = a -n-X.B-n$ .

Clearly, n E ~(A). Then it is shown, by the same proof as in

E2], that the sequence (11 ~ nil )n~ 0 is monotone decreasing,

but for the sake of completeness we will write out the argument.

Following the proof of [21, one has a .A~ n+l = ~ n and, by ap- A 220 Ota and Schmiidgen plying Lemma 3.1 for the formally hyponormal operator ~ .A,

II ~ n+lll ~ I/2(ll ~ nil + l{ ~ n+2ll ) (n~ 0).

Since the sequence { {} ~ nl{ I is uniformly bounded, it follows that it is monotone decreasing. In particular, one has

{{ A-X.B-I~ {{ = {{ X-B-B-I~ II = {l $ 0ll

{{ $ 1 {{ = a -I{{ X'B-I~ {{ .

Let

T = A{ X~(B) be the restriction of A to X~(B). Since X has dense range, it follows that T is densely defined, and T is bounded from below by the above inequality. Since

~(T) = A.X~(B) = X'B~(B) = ~(X), it follows that T has dense range too. Hence from the standard operator theory, T* has a bounded inverse with ~(~) = ~ .

This implies that A* (c T*) has a continuous inverse on ~(~).

Since A is formally hyponormal, it follows that

Xer(A) ~ Ker(A*), and so

~(~) D ~(A)

Clearly A has dense range. Hence, A* has dense range. This means that A* has a continuous inverse with ~(A *) = ~ .

Therefore, A = (A*)* has a bounded (everywhere-defined) inverse. This completes the proof of the theorem.

The following is an immediate consequence of the above theorem. 0ta and Schmfidgen 221

COROLLARY 3.4. Quasi-similar closed formally hyponor-

mal operators have equal spectra.

4. NORMAL OPERATORS

Ne first give a generalization of Douglas' result in

[6] to unbounded operators.

THEOREM 4.1. Let A and B be normal operators. If

A and B are quasi-similar, then A and B are unitarily

equivalent.

Proof. Let A1(resp. B 1) be the closure of 1/2.(A

+ A*) (resp. 1/2-(B + B*)); the real part of A (resp. B) and

let A2 (resp. B 2) be the closure of 1/2L .(A - A*) (resp.

1/2L .(B - B*)); the imaginary part of A (resp. B), where L = r It then follows from the spectral theory for a normal

operator (ex., Theorem 7.32 in [20],or Ch. XII. Exercise 9.10 in

[7]) that all A i, Bi(i =1, 2) are self-adjoint operators

satisfying

AI-A 2 = A2-A 1 , BI"B 2 = B2.B 1

and

A = A 1 + L A 2, B = B 1 + L B 2.

Let X = XAB be a quasi-invertible transformation with

X-A c B.X. It then follows from the Fuglede-Putnam theorem

([12]) that X-A* c B*.X, and so

X.A 1 c B1.X , X.A 2 c B2.X.

Since both of L .I - A i and L -I - B i are (everywhere- defined) boundedly invertible and 222 0ta and Scb.mfidgen

X.(L .I - A i) c (s -I Bi)-X

for i = I, 2, it follows that

X.(L .I - Ai)-1 = (L .I Bi)-l.x

for i = I, 2.

Let X = I X*[ "U be the polar decomposition of X.

Clearly U is unitary. Applying the proof of Lemma 4.1 in

[6] or Theorem 1.6.4 in [13] for bounded normal operators

(L .I - Ai)-i and (L .I - Bi)-l, it is easily seen that they are unitarily equivalent with the common intertwining operator U, namely

U'(~ -I - Ai)-I = (s 'I - Bi)-I'u

for i = 1, 2. Hence, by noticing that U is unitary, one has

U.A = U.(A 1 + L .A 2) = B1-U + L .B2oU

= B-U.

This completes the proof of the theorem.

The following proposition and corollary are motivated by the corresponding to the results related to unbounded repre-

sentations of an algebra in a Hilbert space, see [11] for the

definition and discussion on them.

PROPOSITION 4.2. Let A be a closed densely defined

symmetric operator in a Hilbert space. If A is quasi-similar

to its adjoint A*, then A is self-adjoint.

Proof. Let X be a quasi-invertible operator such

that X.A* c A.X. For ~ ~ ~(A *) with A*~ = ~ ~ , one has

X~ ~ (A) and

L .x~ = X.A*~ = A-X~ . ~ta and Schmiidgen 223

Since the point spectrum of a symmetric operator is real, it fol-

lows that X~ = 0 and, by the injectiveness of X, ~ = 0.

Analogously,

(~(A - ~ "I)) "t" = Ker(A* + L -I) = (0).

Hence the deficiency indices of A are (0, 0). This means that

A is self-adjoint.

REMARK 4.3. In general the above proposition fails if

the closed operator A is merely normal, and so A is unitarily

equivalent to A* ( see [19] for such an example and further in-

formations in case of bounded operators ). In the special case,

however, that a normal operator A is quasi-similar to a self-

adjoint operator, as we will see later (in Corollary 4.6), A

turns out to be self-adjoint.

COROLLARY 4.4. Let A be a densely defined, closed

symmetric operator. If A is quasi-similar to a self-adjoint

operatDr B, then A is also self-adjoint, and moreover A and

B are unitarily equivalent.

Prp_~. By Lemma 2.2 and Proposition 2.4, one has A

and B* ~ A*. It follows from the essentially self-

adjointness of B that A ~ A*. The corollary follows from

Theorem 4.1 and Proposition 4.2.

THEOREM. 4.5. Let A be a closed subnormal normal

operator and B be a self-adjoint operator in a Hilbert space

. If there is a positive, quasi-invertible operator X on such that 224 ~ta and Schmiidgen

X.B c A.X, then, A = B.

Proof. Let N be a normal extension of A on a Hil- bert space ~ with X= ~9~ -u . Let X 0 and B 0 be the ex- tensions of X and B to X as follows:

(X 0) X 0 = w.r.t. ~9~ -L , 0 0

(B O) = (B)~ g_u , and BO(~ ~ ) = B~ $ y .

Then the assumption implies that X0.B 0 c N.X 0 It follows from the Fuglede- Putnam theorem that X0-B 0 C N*.X 0. Hence, one has

Xo.N c Bo.X O.

In particular, X.A c B.X. It follows from Theorem 3.3 and

Lemma 2.8 that theresolvent p (A) of A is equal to p (B).

Moreover, the relations X.B c A.X and X.A c B.X yield

X2.B c X.A.X c B.X 2.

Since B is self-adjoint, one has X.B c B.X, using the stand- ard spectral theory.

Since s 6 p (B) and so L ~ p (A), the inequalities

X(L -I - B) c (~ .I - B)X and X(L .I - A) c (s .I - B)X imply that

X(s -I - A) -1 = (s .I - B)-Ix = X(s .I - B) -1.

Hence (s .I - A) -I = (/. "I - B) -1. This means that A = B.

COROLLARY 4.6. Let A be a closed subnormal operator in a Hilbert space ~ . If A is quasi-similar to a self- adjoint operator in ~ , then A is self-adjoint and is unitarily equivalent to B. (~ta and Schmiidgen 225

Proof. Let N be a normal extension of A and let be a quasi-invertible operator such that X.B c A-X. Let X =

U.P be the polar decomposition of X. Then U is unitary and

P is positive. Since U*AU.P = U*AX ~ U*XB = P.B and U*AU

is subnormal, it follows that U*AU = B.

REFERENCES

1. G. Biriuk and E.A. Coddington, Normal extensions of un- bounded formally normal operators, J. Math. Mech., 12 (1964), 617-638.

2. S. Clary, Equality of quasi-similar hyponomal operators, Proc. Amer. Math. Soc., 53(1975), 88-90.

3. E.A. Coddington, Formally normal operators having no normal extensions, Canad. J. Math., 17(1965), 1030- 1040.

4. J.B. Conway, On quasisimilarity for subnormal operators, Illinois J. Math., 24(1980), 689-702.

5. J.B. Conway, Subnormal operators, Pitman Adv. Pub. Program, Boston-London(1981).

6. R.G. Douglas, On the operator equations S*XT = X and related topics, Acta Sci. Math., 30(1969), 19-32.

7. N. Dunford and J.T. Schwartz, Linear operators, ~ol.II, Wiley-Interscience, New York-London-Sydney-Toronto (1971).

8. B. Fuglede, A eommutativity problems for normal operators, Proc. Nat. Acad. Sci. U.S.A., 36(1950), 35-40.

9. P.R. Halmos, A Hilbert space problem book, Springer- Verlag, Berlin-Heidelberg-New York(1951).

I0. T.B. Hoover, Quasi-similarity of operators, Illinois J. Math., 16(1672), 678-686.

11. S. ~ta, Unbounded representations of a *-algebra on in definite metric space, preprint(1985).

12. C.R. Putnam, On normal operators in Hilbert space, Amer. J. Math., 73(1951), 357-362.

13. C.R. Putnam, Commutation properties of Hilbert space 226 ~ta and Schmfidgen

operators and related topics, Springer-Verlag, Berlin- Heidelberg-New York(1967).

14. K. Schmfidgen, A formally normal operator having no normal extension, Proc. Amer. Math. Soc., 95(1986), 503-504.

15. J. Stochel and F.H. Szafraniec, On normal extensions of unbounded operators I, J. Operator Theory, 14(1985), 31-55.

16. J. Stochel and F.H. Szafraniec, On normal extensions of unbounded operators II, to appear in Acta Sci. Math.

17. M.H. Stone, Linear transformations inHilbert space and their applications to analysis, Amer. Math. Soc. Col- loq. Publ., 15, Providence(1932).

18. B. Sz-Nagy and C. Foias, Harmonic analysis of operators on Hilbert space, North-Holla,d, Amsterdam, American Elsevier, New York-Acad. Kidao-Budapest(1970).

19. J.P. Williams, Operators similar to their adjoints, Proc. Amer. Math. Soc., 20(1969), 121-123.

20. J. Weidmann, Linear operators in Hilbert spaces, Springer-Verlag, Berlin-Heidelberg-New York(1980).

S. Ota K. Schm~dgen Department of Mathematics Section Mathematik Kyushu Institute of Design Karl-Marx-Universitit Fukuoka, 815 Japan Leipzig, 7010 D.D.R.

Submitted: October 26, 1987 Revised: July Ii, 1988 Integral Equations 0378-620X/89/020227-1451.50+0.20/0 and Operator Theory (c) 1989 Birkh~user Verlag, Basel Vol. 12 (1989)

EXTENSIONS OF INTERTWINING RELATIONS

Vlastimil Pt~k and Pavia Vrbov~

Let U 1 and U 2 be two isometries acting respectively on the Hilbert spaces H 1 and H 2 and let M 1 be a Ul-invariant subspace of H I . Let X:M I --+ H 2 be a bounded linear operator intertwining U 2 and the restriction to M 1 of U I ,

X( hIMr = U2X The authors give necessary and sufficient conditions for the existence of a bounded operator Y:H I --+ H 2 which extends X and satis- fies Y U 1 = U 2 y .

In the course of their study of generalized Hankel operators the authors investigated, not long ago, intertwining relations of the form

X T *1 = T 2 X

where T I and T 2 are given contractions acting on Hilbert spaces H I and H 2 respectively. The problem considered [3] was to lift the above identity to a relation of the form

* y Y U 1 = U 2

the U. being the minimal isometric dilations of the corresponding T. . I l It turns out that there is an essential difference between this lifting problem and the classical one of Sarason-Nagy-Foia~. Whereas relations of the form X T 1 = T 2 X may always be lifted to Y U 1 = U 2 Y , in the case considered above (where I'* I is to be dilated to a coisometry U 1 additional condition has to be imposed for the lifting to be possible. This additional requirement assumes the form of a stronger boundedness condition to be satisfied by X . The authors [4] called it R-boundedness: indeed, it is easy to see that any operator Y intertwining U~ and U 2 , 228 Pt~k and Vrbov~

YU~ = U2Y , satisfies the relation Y = P(R2)Y = YP(R I) where the P(Ri) are the orthogonal projections onto the subspaces R.I such that uiIR i is the unitary part in the Wold decomposition of U i . As a compression of an operator Y satisfying Y = P(R2)YP(RI) the operator X clearly satisfies

(Xhl,h2) =(Yhl,h2) = IYP(RI)hI,P(R2)h2)

It follows that

I(Xhl,h2) l ~ IYI IP(RI)hl I IP(R2)h21

and this condition is stronger in general than ordinary boundedness. The referee of [3] called the authors' attention to the fact that an extension problem for shift operators considered by L. B. Page [2] also requires an additional continuity condition in order to possess a solution, while the corresponding problem for coisometries is always solvable. To be more precise: given a coisometry S* on a Hilbert space H , an S*-invariant subspace M C H and an operator T O @ B(M) which commutes with the re- striction S*IM it is possible to show that there exists an extension T of T O to the whole of H which commutes with S* and such that ITI IT01 If S* is replaced by an isometry the analogous extension result is false in general. Indeed, Page considers a unilateral shift S on a Hilbert space H , M an invariant subspace for S and T O a bounded operator on M commuting with SIM . He then shows that operators T O for which an extension is possible satisfy an additional condition in the absence of which the result may fail: he exhibits an example of a two-dimensional shift and an operator T O such that no extension of T O will commute with S , (For a shift of multiplicity one an extension is always possible.) In the same paper Page formulated the conjecture that the necessary condition is also sufficient for the existence of an extension. The con- jecture was subsequently proved in the particular case when S is a finite dimensional shift by C. F. Schubert [5]. The general case remained open. In the present paper the authors consider the more general case of isometrics and replace the commutation relation by a relation intertwining two different isometries. This has another further advantage: by separating the domain and the range spaces of the given operator a deeper insight is Pt~k and Vrbova 229

obtained. We consider two isometries U 1E B(HI), U2EB(H2), a Ul-invariant

subspace M I of H 1 and a bounded operator X:M I + H 2 intertwining UIIM 1 and U 2,

x(u 1 IMI) = hx

The main result is a necessary and sufficient condition for the existence of a bounded operator Y which extends X to the whole of H 1 and intertwines U 1 and U 2 . The condition turns out to be a generali- zation of Page's condition; it is interesting that the condition only refers to the shift parts of the isometries U 1 and U 2 .

I. Notation and preliminaries In the whole paper we use a number of facts from dilation theory. As a general reference for these results as well as for the theory centered around the commutant lifting theorem we suggest the book [I]. Given a Hilbert space H and a bounded linear operator T on H we shall denote by Inv(A,T) and Red(A,T) the smallest T-invariant (resp. T-reducing) subspace of H containing a given subset AC H . If B is a A subspace of a Hilbert space A we denote by PB the orthogonal projection from A onto B . For brevity, we use the symbol P(B) whenever there can be no doubt as to which space A is meant.

If U e B(H) is an isometry we have H = R ~ i where R and Z~t are U-reducing with UIR unitary and UIE -L a unilateral shift. Denote by V ~ B(K) the minimal unitary extension of U so that VIH = U . We shall adopt the following convention concerning orthocomplements. Since the space H appears in the statements of the results as opposed to K , used only in the proofs, we shall understand A~L to be HC) R ; otherwise Z • = ~(L) Z . For each n ~ 0 we shall denote by Pn ~ B(H) the projection operator onto H ~ UnH = Ker U *n = H{~ VnH • , in other words, P = n 1 - unu *n . For h ~ H , we have (i - unu*n)h = h - vnp(H)V*nh = vnp(Hi)V*nh . The relation

P h = vnp(Hi)V*nh n will play an important r$1e in our further work. In particular, it implies the equality of norms 230 Pt~k and Vrbov~

IPnhl = Ip(Hi)v*nhl

for all h~H . The following technical lemma will play an important r$1e in our considerations. i.i Lemma. Let MC H be invariant with respect to an isometry U . Set

A = (P(H • Red (M,V))-

Then 1 o Red (M,U)~ R • = Inv (A,V) G A , 2 ~ the operator T = P(H• coincides with the compression of V to A , i.e. p(Hi)VIA = P(A)VIA and its minimal isometric dilation may be identified with V I Inv(A,v) Proof. The proof of the first part of the proposition is based on some iden- tities relating subspaces of K and H which may be stated in a somewhat greater generality. For simplicity, let us write P and Q for P(H) and P(H • . Consider a V-invariant subspace B C K and let us show that

Inv (QB,V) Q QB = P(Inv (QB,V)) .

This equality, in its turn, will be a consequence of the inclusion

Q Inv (QB,V) C (QB) (i)

Indeed, given a y ~ Inv (QB,V) Q QB we have y • QB and it follows from (i) that Qy ~ (QB)- Hence IQyl 2 = (Qy,y) = 0 so that y = Py + Qy = Py . To prove inclusion (i) observe that VHC H so that VP = PVP and QV = QVQ . It follows that

QV n = (QVQ) n = QVnQ

for every n . To prove the desired inclusion it obviously suffices to show that QVnQBC QB for every n ; this, however, follows from the identity QV n = QVnQ .

Consider now the case B = Red (M,V) . We have then A = (QB)- and Inv (A,V) @ A = Inv (QB,V) G QB = P(Inv (QB,V)) Pt~k and Vrbov~ 231

Now let us prove the inclusion

PInv (QB,V) C Red (M,U) .

Since pv*km = u*km for all m E M and k ~ 0 we have

PBC Red (M,U) .

Furthermore,

pvkQB = pvk(I - P)BC pvkB + pvkpBC PB + ukpB Red (M,U) + U k Red (M,U) ~ Red (M,U) .

We have thus proved

PInv (QB,V) C Red (M,U) .

Since Range Q • R and R is V-reducing, we have Inv (A,V) • R and, consequently,

PInv (A,V) C P(KQR) = R • , the orthocomplement of R in H . It remains to prove the inclusion Red (M,U)~ R• Inv (A,V)QA . Since Red (M,U) I~ R• H and H • A we have Red (M,U)(~ R• HC A • To prove the inclusion Red (M,U)~ R• Inv (A,V) we observe first that Red (M,U)~ R• P(R • Red (M,U) . Since P(R• = lira P h = n lim VnQV*nh for every h ~ H it suffices to consider elements of the form VnQV*nUPU*qm , m C M . Using the relation V*Q = QV*Q we obtain

VnQV*nuPu*q m = VnQV*n-P(1 _ Q)v*qm = vnQv*n-p+qm _ vPQV*qm

E Inv (A,V) .

This proves the inclusion and completes the proof of 1 ~ Using the identity QV n = QVnQ = (QV) n we deduce that

QVnQB = QVnBC QB~A so that QvnAc A and

Qvna = p(A)QVna = p(A)vna for a E A and n ~ 0 . Furthermore, 232 Pt~k and Vrbov~

(p(A)VP(A)) n = (p(A)v)np(A) = (QV)np(A) = Qvnp(A) = P(A)Vnp(A)

so that V I Inv (A,V) is the minimal isometric dilation of P(A)V I A . N

The following proposition will he useful in the sequel. 1.2 Proposition. Let U 1E B(HI) and U 2 E B(H 2) be two isometries , M i a subspace of H 1 invariant with respect to U 1 . For n ~ 0 , let Pln ~ B(HI) and P2n E B(H 2) be the projections in H I and H 2 defined by the relations

pl i n ,n p2 I - U nU*n n = - UIUI ' n = -2~2

Suppose X: M 1 -~ H 2 satisfies the relation

x(h i 1) = u2x (2)

Then the following assertions are equivalent

l~ IP Xml IPn ml

for all m ~ M I , n >~ 0 .

2 ~ IP(H~I ~k v2kXmkl--< ~IP(HI) k~ Vlkmkl

for all finite sequence ml,m 2 ..... m k ~ M I

Proof. Take a finite sequence m I ..... m k ~ M 1 . Then

n n PIH~) k-1_[ V 2kXmk = P(H~) V ,n2 k__LlV2v ,n-k. ~mk =

,n (V2P(H~n ,n n-k = v,np2 ~,n~2= ~ .n-k V 2 :iv2 2 n = = "2 ~nAk~lUl mk "

Suppose condition 1 ~ holds. Then

n n n

n n alVIP(HI)VI k=~IV1 mkl Pt~k and Vrbov~ 233

Now suppose 2 ~ satisfied. We have then, for m 6 M 1 , n ~ 0 ,

Ip2Xml n _t ,n = < a = IV2P(H2) V2 Xm I Ip(H~)V2nxm I = ~n~1,,n vl m[ =

~ IVlP [H• = a Iplml i)~i m

The following two examples suggested by the referee illustrate the results of this section as well as Theorems 3.2 and 3.4. Example i. Let U I be the multiplication by z on H I = L 2, and let U 2 be its restriction to the Hardy space H 2 = H 2 . Set M I = H2~ L 2 . If we denote by X the identity mapping of M I into H 2 , i.e. X : f--+ f for f ~ H 2 then obviously X(UIIM1) = U2X . Since Pln= 0 for all n a 0 the operator X does not satisfy condition i ~ of 1.2. This shows that the intertwining relation (2) alone does not imply condition i ~ Observe that X cannot be extended to a mapping Y of L 2 into H 2 for which YU I = U2Y . Indeed, only the zero operator can intertwine a bilateral and a unilateral shift: the relation YU~ = U~Y shows that the range of Y is contained in the inter- section /~U~ H 2 (n ~ O) and this is the zero snbspace.

A little more challenging is the following Example 2. Denote by H I = H2~ L 2 and let us define U I ~ B(HI) as the direct sum of multiplications by z on H 2 and L 2 ; thus U 1 is a direct sum of a unilateral and a bilateral shift. Then R 1 = (0) O L2, A~I = H20 (0) and the minimal unitary extension V 1 of U I is the direct sum of two bi- lateral shifts on L2(~ L 2 defined by VI: (f,g) --+ (zf,zg) . Consider the following Ul-invariant subspace M 1 = {(f,f) : f 6 H 2} . It is easy to show that Inv (~,U~) = {(h,f) : feL 2 and P+f = h], Red (M1,U1) = H 1 , Red (~,U I) /~I =~I =H2~ C0) , Red (MI,VI)= {(f,f) : fEL 2] , P(HI) Red (~,Vl) = H ! ~ (0) and Inv ~(H~) Red (MI,VI)) = L 2 ~ (0).

Here, as usual, P+ denotes the orthogonal projection of L 2 onto H 2 and H 2 = L2~H 2 .

This illustrates the identity 1 ~ in Lemma i.i. Furthermore, if U 2 6 B(L 2) is the operator of multiplication by z , and if X : M 1 --+ e 2 is defined by X : (f,f) --+ f then X(UIIM I) = U2X and IxI = i/~ . 234 Pt~k and Vrbov~

It is easy to show that the set of all extensions of X to H 1 which intertwine U 1 and U 2 consists of all operators of the form Y~ : (f,g) --+ ~f + (i _ ~)g with some ~ ~ g ~ . The choice ~ ~ 1/2 yields an ex- tension of X whose norm does not exceed that of X .

2. Operators intertwining parts of coisometries We deal first with the problem of extending operators intertwining restrictions of two coisometries. In this case the existence of extensions may be deduced directly from the commutant lifting theorem.

2.1 Theorem. S~ppose U 1 ~ B(HI) and U 2 C B(H 2) are isometries and let a subspace MIC H 1 be given such that U~MIC M 1 Suppose that the operator X E B(MI,H2) satisfies

xIu I MII ,

then there exists a Y ~ B(HI,H2) which extends x and intertwines u 1 and

U 2 , Y I MI ~Y , *y Y U I = U 2 ,

IY1 = Ixl

Proof. Since X(U~ I M 1) = U~X , its adjoint X*: H 2 -~ M 1 satisfies P(M1)UIX* = X*U 2 . It is not difficult to see that the minimal isometric dilation of the contraction P(MI)U 1 I M 1 may be identified with U I I Inv (MI,UI) - To see that, it suffices to observe that UIM~C M S whence P(MI)UIP(M~) = 0 and, consequently, P(MI)UIP(MI) = (P(MI)UIP(MI))n For brevity, set I = Inv (MI,UI) According to the commutant lifting theorem there exists a Z @ B(H2,/) such that UIZ = eu 2 , Iz[ = Ix*[ and PIM1Z = X*

Now define Y: H I --+ H 2 as the adjoint of Z considered as an element of B(H2,H 1) . We have then

YU~ = U~Y and IYI = IZ*l = IZl = IX*l = [XI Pt~k and Vrbov~ 235

For h I ~ M 1 and k 2 6 H 2 we have

(Yhl,k2) = (hl,Zk2) = (h1,P~iZk 2) = (hl,X*k2) = (Xhl,k2)

It follows that Y is an extension of X .

3. The general case of two isometries The case of two arbitrary isometries requires more delicate con- siderations. We begin by considering two particular cases - each of them requires a different treatment. 3.1 Theorem. Let H 1 and H 2 be two Hilbert spaces and U 1 , U 2 two isometries acting on H 1 and H 2 respectively. Suppose we are given a U]-invariant subspace M l C H 1 and a bounded linear operator X: M 1 --+ H 2 such that X(U 1 r M I) o u2x

Then the following assertions are equivalent 1 ~ there exists a bounded linear operator Y: H l -~ H 2 such that

Y U 1 = U 2 Y ,

YJM I = P(R )x ,

Iyl ~a ,

2 ~ the operator X satisfies the inequality aIPmJ for all m C M 1 and all n ~ 0 .

Proof. It is easy to see that 1 ~ implies 2 ~ (see also [2]). Indeed, if n n YU 1 = U2Y then YUIH I_C U2H 2 for n > 0 . This implies that Y* Range p2 n y,p2 = Y*(H 2 e U2H 2) C (H 1 • UIH I) = Range pl in other words n ~ n

= ply,p2 , or equivalently, p2yp1 = p2y . This implies, for m ~ M 1 and n n n n n n> 0 , p2Xmn = P~P(R~)Xm = p2ymn = P2ypImnn " We have thus

Ip2Xml ~ IP2YI IPnlml _-< IYl Iplml for all m E M 1 and n ~ 0 . This proves 2 ~ . 236 Pt~k and Vrbov~

Conversely, assume that 2 ~ is satisfied. Denote by Z 0 the operator Z 0 : [P(H~) Red (MI,VI))- --+ H •2 defined by the relation

ZoP(Hi)V~km = P(H~)v~kxm

for all m C M I and all k ~ 0 . The existence and continuity of Z 0 is guaranteed by assumption 2 ~ and Proposition 1.2; in particular, IZ01 ~ ~ . Let us introduce two contractions T 1 and T 2 defined as follows

T I = P(AI)V IIA I = P(Hi)V 11A1

where A 1 = P(Hi) Red (MI,V I) and

T 2 = e(H~)V21H i

and let us show that Z0T 1 = T2Z 0 . It suffices to prove this for elements of the form P(Ht)V~km with m ~ M 1 and k ~ 0 .

Indeed,

ZoTIP(H~)v~km = ZoP(H~)VIP(Hf)v~km = ZoP(H~)VIV~km = Z~P(H•u " i'-i *k"ulm

: p (Hi~V 2"-2 *k~' Avlm : P(H~)v~ku2xm : P (H~)V2(P(H2) + P(H~))V~ kXm

= P(HI)V2P(H~)v~kxm = T2ZoP(H~)v~km

According to the commutant lifting theorem and Lemma i.i there exists an operator Z : Inv(Ai,Vl) --+ Inv(H~,V2) such that

ZV 1 ] Inv(Ai,V I) = V2Z ,

z0=P(s )z I A 1 ,

Z(Inv(AI,V 1) ~ A I) C Inv(H~,V 2) @ H i

and

Set Y = ZP(D 1) I H I where D I = Inv(Ai,V I) ~J I . According to Lemma i.I we have Inv(H~,V2) @ H E = H 2 ~ R 2 = B~ so that Range Y C H 2 and D I = Red (M1,UI)(~ R~ so that D I is reducing for U I . It follows that Pt~k and Vrbov* 237

YU I = ZP(DI)U I = ZUIP(DI) = ZVIP(DI) = V2ZP(DI) = V2Y = U2Y

To complete the proof, let us show that Y is an extension of P(R~)X . Since Y = P(R~)Y it will be sufficient to prove that, for each m ~ M I and each positive integer n , p2ymn = P2Xm " We have observed already that the relation YU 1 = U2Y implies p2y = p2yp1nn for all n . It follows that, for m E M 1 ,

p2ymn = Pn2yplm ~ V2 P (H2)• V2,n ZP (D1) Pnlm "

Since plmn ~ Red(MI'UI) (~ Rt ~ D1 the last expression equals

n • ~n i V2P(H2)V ~ ZPnm = V2P(H2)Vn s 2,n ZVIP(HI)Vn • ,n1 m .

Observing that P(Hi)vlnm E A 1 we may simpl$fy the above expres- sion to

V2(P(H~)Z) i ,n n • ,n n • ,n P(HI)V 2 m = V2ZoP(HI)V 1 m = V2P(H2)V 2 Xm -- p2Xmn "

Let us state explicitly two particular cases of the preceding theorem. If U 2 is a shift operator then U 2 is an isometry for which R~ = H 2 so that, in the preceding theorem, Y will be an extension of X .

3.2 Theorem. Let H 1 and ~2 be two Hilbert spaces, U I an isometry on H 1 and S 2 a shift operator on H 2 (thus S~S 2 = 1 and s~nx -~ 0 for every x C H 2 ). Let M 1 be a subspace of H 1 invariant with respect to U I . Let X : M 1 -~ H 2 be a bounded linear operator such that x(h I 11 h x

Then the following assertions are equivalent i ~ there exists a bounded Zinear operator Y : H I -~ H 2 which extends X and satisfies YU 1 = S2Y ,

IYI ~a .

2 ~ the operator X satisfies the inequality

for a~ m ~ M~ and all n > 0 . 238 Pt~k and Vrbov~

Specializing further we are able to establish the validity of the conjecture stated in the 1971 paper [2] of L. B. Page. 3.3 Theorem. Let H be a Hilbert space and M a subspace invariant with respect to a unilateral shift S on H . Let X be a bounded operator defined on M which commutes with S Then these are equivalent

1 ~ there exists an extension Y of X to the whole of H which also con~nutes with S and IYl ~ ~ ,

2 ~ IPnXm I ~ aIPnm ]

for all m s M and all n ~ 0 (here P = 1 - sns *n ). n 3.4 Theorem. Let H I and H 2 be two Hilbert spaces, U 1 an isometry on H 1 , U 2 a unitary operator acting on H 2 . Suppose we are given a U l- -invariant subspace MIC H I and a bounded operator x : M 1 --+ H 2 such that xIu I I MII = u2x

Then there exists a bounded operator Y : H I -+ H 2 such that

YU 1 = U2Y ,

YIM 1 = X , IYl : Ixl

Proof. Let V 1 ~ B(KI) be the minimal unitary extension of the isometry U 1 . Denote by W the compression to M 1 of U *1 and let us show that V I* I Red(M I,V 1) is the minimal isometric dilation of the coisomtry W

Indeed,

P('l)V~ k I M i = (vk I Mi)* : (uik I Mi)* = (U I I MI )k* : (U 1 IMi)*k: W k for all k ~ 0 . Since kYOV~ kMl : Red(MI,V I) it follows that V *I I Red(MI'VI) is the minimal isometric dilation of W . Since U 2 is unitary we have

x : .~x(~ I L MI) : ~x.* . According to Lemma 2.8 of [4] there exists a unique operator Pt~k and Vrbov~ 239

X 0 : Red(M1,Vl) --+ H 2 such that

X 0 = U~X(V~ I Red(MI'VI))* = U~XV 1 I Red(MI,VI) '

X 0 I M 1 = X ,

lXol = Ixl.

NOW set Y = XoP(RedfMI,VI) ) ] H 1 . Abbreviating P(Red(MI,VI) ) to R , we have then, for each h C H 1 ,

YUIh = XoRUlh = XoRVIh = XoVIRh = U2(U~XoV1)Rh = U2XoRh = U2Yh

At the same time, for m 6 M I ,

Ym = XoRm = X0m = Xm .

The proof is complete.

Theorems 3.1 and 3.4 make it possible to describe fully the general situation. 3.5 Theorem. Let H 1 and H 2 be two Hilbert spaces, U 1 and u 2 two isometries acting on H I and H 2 respectively. Let MICH 1 be a sub- space invariant with respect to U 1 . Let X : M 1 --+ H 2 be a bounded linear operator such that x(u1 i MI) = u2x

Then X may be extended to an operator Y : H 1 ~ H 2 satisfying YU 1 = U2Y if and only if there exists a positive a such that

for all m ~ M 1 and n ~ 0 . More precisely: these are equivalent 1 ~ there exists an operator y : H 1 -+ H 2 such that

YU I = U2Y ,

YIM I = X ,

IP(R2)Y I ~ IP(R2)X I and IP(R~)YI $ a ,

2 ~ for every m E M I and every n ~ 0 , 240 Pt~k and Vrbov~

Proof. Suppose first condition 2 ~ satisfied. Write X in the form X = P(R2)X + P(R~)X and verify that each summand satisfies the same intertwining relation. Then apply Theorems 3.1 and 3.4; we obtain an extension Y1 of P(R2)X and an extension Y2 of P(R~)X such that IYI[ = [P(R2)X [ and [Y21 $ m , both fulfilling the intertwining relation on the whole of H I . The operator Y = Y1 + Y2 satisfies P(E2)Y = YI ' and P(R~)Y = Y2 and satisfies thus the requirements of the theorem.

The authors acknowledge a debt of gratitude to the referee for a number of comments which have contributed to the improvement of the presentation.

References

[i] B. Sz.-Nagy and C. Foia~, Harmonic analysis of operators on Hilbert space, Akad~miai Kiad~-Budapest, North-Holland Publi- shing Comp.-Amsterdam-London, 1970.

[2] L. B. Page, Operators that commute with a unilateral shift on an invariant subspace, Pac. J. Math. 36 (1971), 787-794.

[3] V. Pt~k and P. Vrbov~, Lifting intertwining relations, Intt Eq. Operator Theory ii (1988), 129-147.

[4] V. Pt~k and P. Vrbov~, Operators of Toeplitz and Hankel type, Acta Sci. Szeged (in print).

[5] C. F. Schubert, On a conjecture of L. B. Page, Pac. J. Math. 42 (1972), 733-737.

Institute of Mathematics, Czechoslovak Academy of Sciences Zitng 25 115 67 Praha i, Czechoslovakia

Submitted; April i0, 1988 Revised: July 15, 1988 Integral Equations 0378-620X/89/020241-3951.50+0.20/0 and Operator Theory (c) 1989 Birkh~user Verlag, Basel Vol. 12 (1989)

On the Almost Unperturbed SchrSdlnger pair of Operators

Daoxing Xia*

The pair of unbounded self-adjoint operators {U, V} satisfying the condition that i[V, V] c Z + D, where D is in the trace class, is investigated. Some theorems on symbols, the integro-differential model, the principal distribution and the determining function of this {U, V} are established.

1. Introduction. The basic operators in the quantum mechanics is the SchrSdinger pair of operators q,p on L2([R) defined by

(qf)(x) = xf(x) for f E P(q) = (f E L2([R) : (-)f(-) E L2(~)} and

(pf)(x) = id f(x) for f E P(p) = {f E Lz(IR) : f is absolutely continuous on IR and ft E L2(IR)}. This SchrSdinger pair of operators satisfies the Heisenberg commutation relation

ilq, p] C I.

However, this commutation relation can not characterize this pair due to the un- boundedness of {q,p} (cf. [10]). The Weyl commutation relation

eiqseipte-iqse -ip~ = eiatI, for s,t E does characterize this SchrSdinger pair of operators by Von Neumann Theorem (cf [13] or [14]). In the case of a pair of unbounded self-adjoint operators, Von Neumann theorem may be written as follows.

* Supported in part by NSF grant DMS-8700048 242 Xia

THEOREM. Let {U,V} be a pair of self-adjoint operators on ~ satisfying the Weyl commutation relation

eiUseiVte-iUse -ivt = eiStI, for s,t 6 gR.

Then there exists an orthogonal decomposition of

g = @g,, ot such that g~, reduces {U, V} and

WaU[~ W~ 1 = q, W~VI~ W~ I = p where W~ is a unitary operator from Ha onto L2(IR). The Weyl commutation relation is equivalent to the following commutation relation of resolvents

i[(U - AI) -1, (V - ~I)-11 = (U - AI)-I(V - lzI)-z(U - AI) -1 for A,# 6 ~\[R or

i[(u - At) (v - +a)-q = (v - - A1)-'(v - .I) for A, # e C\IR. In this paper, a kind of trace class "perturbation" of these commutation relations is studied. Let {U,V} be a pair of self-adjoint operators on ~, and a 6 IR. This pair of self-adjoint operators {U,V} is said to be art almost unperturbed Sehrdidinger pair of operators with parameter a # 0 if there is a trace class operator D such that i[U, V]f = af + Dr, for f 6 M, (1) where M c P(U)nP(V) is a linear manifold dense in g satisfying UM C P(V), VM c D(U), and M = (U- zI)-IP(V) or M = (V - zI)-lP(U) for some z 6 ~\IR. For a =/- 0, in order to investigate this {U, V} we only have to consider the ease a = 1 and the general case can be treated by a simple transfor- mation. If a = 0, then this pair is said to be almost commuting. There is a deep theory, especially Pincus principal function, on almost commuting operators (cf. [1-4], [6], [8-9], [15]). In this paper, some of them will be generalized to the case of a#O. Xia 243

In the Lemma 1, it is proved that the condition (1) is equivalent to the following i[(U - AI) -1 , (V - t*I) -1] = (U - ~I)-] (V - l~I) -1 (aI + D)(V - ~I)-1 (V - #I)-1 for some or all )% # E ~\IR. The condition (1) also implies that

eitr~eiVte-iU~e -ivt- ei~tl E [1()l), for s,t in R.

That means {eikeiWe ivt : k, s, t E IR} is an "almost" Weyl group of operators in the sense of [16] (cf. w of this paper). In w the existence of the positive and negative symbols w• U) which is defined by

S -- lim ei~UeitV e -iav--iastPe ac~.IU~ ) = eitO,e(v;U)tpa e(U), 8--~ • oo where P,~c(U) is the projection to the absolutely continuous spectral subspace for U, is proved. As a matter of fact, these symbols are related to the wave operators. Besides, by means of Von Neumann theorem, it is proved that w_(V; U) is unitarily equivalent to cv+(V; U) if the parameter a r O. In w a singular integro-differential model for the almost unperturbed SchrS- dinger pair of operator with parameter a r 0 is given under certain conditions. The model is (Uf)(m) = mf(m) and a(x)*J f a(s)f(s)ds (vf)(~) = i~ f(~) + 2~---7- ~- (, + i0) where o~(-) is a strongly measurable bounded operator-valued function and J is a sign operator. In w the principal distribution G(., -) for an almost unperturbed Schr6dinger pair of operators {U, V} is defined as the Fourier transform of the function

of s and t. A trace formula

tr(iIC(U, V), r V)] - aJa(r r V)) 244 Xia is given, where r V)is defined by the Weyl-ordered operational calculus, and

i ff i _~ _ Ja(r ~b)(x,y) = 27ra[a I jj (r ri)r - r ri)r ~,(~ )(" Y) d~drl for a -~ 0. Define

20(r162 = 1%J~(r162 0(r162 o(~,~)

Therefore this trace formula is a generalization of the trace formula for the case of a--0. In w assume that the operator D in (1) is of rank one and the only subspace which reduces U and V and contains D)4 is the space )4 itself. Then it is proved that the determining function

E(A,/z) = det(I- iD(U- AI)-I(V - #I)-1), A,# e C\IR is a complete unitary invariant for the pair of almost unperturbed SchrSdinger operators. Under certain conditions, it is proved that the determining function is determined by the principal distribution. In w a result on the finite rank perturbation of Weyl commutation relation and some generation of Von Neumman theorem is recorded. Another paper is preparing for the almost unperturbed SchrSdinger n-tuple of operators.

2. Commutation relations. In this section, we will study the relation between different types of commutation relations.

LEMMA 1. Let U and V be self-adjoint operators on )4 and C E ~.()4). Then the following statements are equivalent. (i) There are ~o and lto in C\~ such that

[(U-Aol)-*, (V-pol) -I] = (U-Aol)-1 (V-#oI)-'C(V-#oI)(U-AoI),-I (2) (ii) For every )~ and Iz in C\rR, [(U-~/)-I, (V-~])-I] = (U-,~I) -I(V-]z/)-IC(V-lz/) -I(U-,~I)-I, (3) Xia 245

(iii) There are Ao and #o in C/R such that

[(V-~oZ) -~, (V-uoZ) -~] = (V-uoZ)-~(V_~oZ)-~c(V_~oZ)-~(V_~oZ)-~, (4) (iv) For every A and # in ~\~,

[(U -- ~I) -1 , (V -/AI)-1] = (V -/.(/)-1 (U - ~1)-1 C(U - .~i)-1 (V -/~1)-1, (s) and (v) There is a linear manifold M C P(U) n P(V) satisfying the conditions that M is dense in g,

VM c v(v), VM C D(U),

there is a Ao E C\IR such that

(U - Aol)M = D(V) (6)

or there is a #o E C\~ such that

(V - #oI)M = P(U), (7)

Besides, [u, v]~ = c~, for ~ e M. (8)

PROOF. In the case (v), if (6) is satisfied, then this condition (v) will be denoted by (vl), otherwise, (if (7) is satisfied) by (v2). (i) ~ (v2). Assume that (i) is satisfied. Let M = (V - #oI)-lP(U). Take any r E O(U). Let y = (U - AoI)r From (2), it follows that

(u - aol)-~(v - ~oZ)-',7 - (v - ~oZ)-'r = (u - Aol)-1 (v - #oI)-Ic(v -/2o1)-I(~.

Therefore (V - #oI)-ar = (U - AoI)-I ~ (9) where ~ = (V - #oI) -I (r/- C(V - #oi)-Ir So, we have (V - #oi)-1r E P(U). Hence M C D(U). It is obvious that M C D(V) and is dense in D(V). Thus M is dense in )/. 246 Xia

On the otherhand, VM c P(U), since (V -I~oI)M = P(U) and M C P(U). From (9}, we have

(v - ~ol)(v - uoI)-Ir = (v - uoO -1(n - c(v - uol)-1r (io)

Therefore (U - AoI)M c P(V). Thus UM C P(V). Let r (V-/~oI)-1r then ~ = (U- AoI)(V- #oi)r and (10) implies that

[(v - ~o~), (v - uoI)]r = cr which proves (v2). (v2) =~ (iv). Let A e r Then from (7), we have

[(U - hi), (V - ~oI)]s" = C~', f e M.

It implies that

f = (U - AI)-' (V - ~ol)-i[(U - M)(V - #oI)q - Cr (11)

For every r e ~, (U - AI)-1r e P(U), therefore

q -- (V - I~oI)-l(U - AI)-1r e M. (12)

Substituting ~" in (11) by (12), we have

I(U-AI),-' (V-~oI) -1] = (U-AI)-I(V-#oI)-aC(V-#oI)-'(U-AI) -a (13) for A 6 C\IR. Notice that if A,B and C are in/~()/) and satisfy the conditions that

1 e p(BAC), 1 e p(CBA) (14) and AB - BA = ABCBA (is) then AB - BA = BACAB. (16) In fact, (14) and BA(I - CBA) = (I - BAC)BA imply that

BA(I - CBA) -1 = (I - BAC)-I BA.

From (15), we have AB(I- CBA) --- BA. Xia 247

Hence AB = BA(I - CBA) -1 = (I - BAC)-IBA. Thus (I - BAC)AB = BA which proves (16). Let A = (U - ),I) -1 and B = (V - ~oI) -1. If [I),[ is sufficiently large, then [[(U- AI)-I[[ __ I/A[-1 is sufficiently small. Hence (14) is satisfied. It is obvious that (13) implies (15). From (16), it follows that

[(U-)il) -I, (V-/zo/) -I] = (V -~ol)-I(u-)~I)-Ic(u- ~l)-l(v -]2oi) -I.

Thus the condition (i) of this theorem for V, U, -C is satisfied for/~o and A with sufficiently large []A[. Therefore the condition (v) of this theorem is satisfied for V, U,-C and M = (U- AI)-lp(V) for sufficiently large [IA[. Hence (5) is satisfied for any/~ E ~\IR and A E ~\IR with sufficiently large [IA[. But both sides of (5) are analytic functions of A and/z, so they must be equal for all A E q~\lR and # E ~\IR. By the argument we used before, it is easy to prove (ii) and (iv) are equiv- alent. It is obvious that (ii) implies (i). Thus (i), (ii), (iv) and (v2) are equivalent. Similarly, we can prove that (ii), (iii), (iv) and (vl) are equivalent. Lemma is proved.

LEMMA 2. Let U and V be self-adjoint operators satisfying the condition of Lemma 1. Then

eiaUVe -i*u = V + i fO '19eirUCe-i~Udr (17) and t eitVUe-i~v = U - i fO ei~V ce-irV dr. (18)

PROOF. Let C(s) = ifo eirUCe-irUdr, then C(s) is self-adjoint. We have to prove that d' (v - = (v + c(,) - (19) for ft E ~\IR. Let U = (V - pI)-lP(U). If r E D(U), then e-i'u~b 6 P(U) and

(v - M)-le- 'v r e M c P(U), 248 Xia by (7) and (v2). Therefore

~se~V (V - #I)- l e-isu r = ieiSU[u, (v - #I)-l]e-iSu r

Take ~ = (V - #I)-le-isur in (8). Then

Ue-i'u r - (V - #I)U(V - #I)-l e-isu r -- C(V - #I)e-i'v r by (8). Hence

[U) (V -- •l)-lle-iau r = -(V - ~I)-lC(V - ~I)-le-isu(~.

Thus

~ed isu( V _ #l)_le_isur = _ieisU(v _ #I)_IC(V _ #I)_le_iSvr for r e P(U), or

K (201 for r E 19(U) and hence for r (5 ~. Let

f(8) -----eisU (v -- t~l)-l e -isU and Q(r) = -ieW'Ce-iUL then (20) is equivalent to the operator identity y(s) = L"I(r)Q(r)I(r)dr + (V - .I)-~. (21) On the other-hand, let h(s) = (V + C(8) - .I)-L then

d h(s) = h(s)Q(s)h(a), since ~C(s)=-Q(s). Therefore

h(s) = h(r)Q(r)h(f)dr + (V - #I) -1. (22) Xia 249

Let C[0,So] be the Banach space of all strongly continuous f()/)-valued functions g(.) on [0, So] with norm Ilall = O

liAr - Ahll <__~o]lCllllf - hll(llfll + Ith/I), since IIQ(s)ll = IlCll. From (21) and (22), f and h are fixed points of A. Therefore

IIf - hll _< ~ollCllllf - hll(llfll + Ilhll).

It is obvious that Ilftl < 1/IX~I and Ilhll -< 1/I-r~,l - Thus f = h if ~011Cll < IZ~l/~ But both sides of (10) are analytic functions of # 6 ~\[R. Hence (19) holds for every # C ~\[R. From (19), it is easy to see that e i~v maps D(V) to P(V + C(s)) = D(V) and hence (17) holds.

COROLLARY 1. If {U, V} is an almost unperturbed Sehrgdinger pair of operators with parameter a, then eisU eitV e-isUe-itV e-iast -- I 6 s ()l) (23) and IleisU eitV e-i, Ue-itV e-iast _ Ill 1 < min(lsl, ItI)lIDl[1 (24) for s, t 6 R, where D D i[U, V] - aI.

PROOF. By (17), ei~VVe -isv = V + D(s) + as where D(s) = f eirU De-iru dr. Therefore eisU eitV e-isU e-itV e-aist __ eit(V+D(s)) e-itv" (25) It is obvious that

deit(VWD(S))e-itvzl = eit(V+D('))iD(8)e-itv ~ 250 Xia

for r/E P(V). Hence

eit(V+D("))e -itv -- I = iei~(v+D(~))D(s)e-i~V dr 6 s (26)

which proves (23). From (26), we have

Ile"(V+D('))e"~ --Ilia _ Itl IIDIla

i.e. the left-hand side of (24) is less or equal to It[ [[Dllt. Similarly, we can prove that

IIe"ve-i~ve-"vei've-~'~ - Ill -< I~t IlDlla Thus (24) is proved.

3. Symbols If A is a self-adjoint operator on 94 with spectral resolution

A = f adE(a),

then the closure of the set of all vectors f in 94 for which IlE(~)ftl 2 is an absolutely continuous function of ,~ is denoted by 94ae(A). It is well known that 94~(A) is a subspace of 94 and reduces A. The projection from 94 to 94~c(A) is denoted by Pat(A).

THEOREM 1. If {U,V} is an almost unperturbed SchrJdinger pair of operators with parameter a ~ O, then the strong limits

a - llm eiSVe itv e-iStre-iastPa~ (U) = e i~• (v;V)tp~r (U) (27)

for t E IR and

a-- li m eiWeiSUe-itVeia~tp.ac~ "v" J = eiW~(O;V)apae(V) (28) t--.-/-oo

for s 6 [R exist, where w• (V; U) and w ~(U; V) are sel[-adjoint operators in 94at(U) and 94at(V) respectively. We will call w+(U; V) and to_(U; V), respectively, the positive and negative symbols of U with respect to V and similarly, w+ (V; U) and w_ (V; U), respectively, the positive and negative symbols of V with respective to U. Xia 251

:PROOF. By Lemma 2, we have V (s) = ei"U (v -- saI)e -i~v = V + f eirll De-irU dr, (29) where D D i[U,V] - aI. Hence V(s) - V E s By a theorem in [5], [7] or [12] the generalized wave operators

fl • (V (s), V) = s - lim eitV(')e-itVpac (V) t---,• = s -- lim eiaVe itv e-iatre-itv e-iastPac (V) (30) ~--*• exist and eomplete. Therefore the strong limits in (28)

s -- limeitVei~VeitVeia'tpac(V ) = Ei(,s) = ei'V~+(V(--s) V) (31) t---,• exist and Ran E+(s) = eiV'xa~(V(-s)). (32) Let the spectral resolutions of V and V(s) be

f V = J tdF(t) and V(s) = f tdF,(t) respectively. From (29) we have

""~ao(V(-*)) = ~,~(V).

Hence Ran ~• = Uoo(V) and -=• are unitary operators from )lac(V) onto gac(V)- On the otherhand, it is easy to see that F.• are strongly measurable functions of, E IR and

Therefore E• ~ E ]R are one parameter groups of unitary operators on )/~c(Y). By Stone's theorem there are self-adjoint operators wi(U,V) on )/~r such that "~+(~) [uoo(v) = e i~ (v,v), which proves (28). Similarly, we may prove (27). 252 Xia

COROLLARY 2. If {U,V} is an almost unperturbed Sehr6dinger pair o[ operators with parameter a # O, then there are unitary operators S(U) and S(V) on g~(U) and )ta~(V) respectively satisfying

s(u)ul~~ -~ = vl~,or (33) s(u)~_(v; u)s(u) -~ = ~+(y; u), (34) s(v)vl~.o(v)s(v) -~ = Vl~~ (35) and s(v)~_(v;v)s(v) -~ =~+(u;v). (36)

PROOF. if s in (27) is replaced by s + b, then we have

eibU e iw~" (V;U)te-ibU e-iabtpae (U) = e iw'~ (v'u)t_Pac (U).

Therefore {Ulu.c (v), w+ (V; U)} and {Ub~.c (v),w- (V; U)} are pairs of SchrSdinger operators on )tat(U). By Von Neumann's theorem, there are sets of subspaces {)4~, a E `4• such that u~(u) = r~e ~. aE.~ • such that {UI;~,w• are irreducible in )1~. and the multiplicities of UI;/~ are 1. Therefore the multiplicity of U in g~(U) equals the cardinal number n + of `4+ and equals the cardinal number n- of .4- as well. Thus n + -- n-. Hence there is a one-to-one correspondence between `4+ and `4-. We may assume that `4+ = `4- = `4. Therefore there exists a unitary operator S~ from )I~- onto )~+ such that

S~Ulu=Sg 1 = U[u+ s~_(v; U)l~:S:' = 0~+(v; u)l~:.

Let S(U) = $~ S~, then S(U) is a unitary operator form ~ac(U) onto ~[ae(U) satisfying (33) and (34).

4. Integro-differential model If /) is a Hilbert space, then the Hilbert-space of all strongly measurable and square integrable P-valued functions f on IR with I[fll = (f Ilf(z)ll~dz) 1/2 is denoted by L2(IR, P). Xia 253

THEOREM 2. Let {U, V} be an almost unperturbed SchrJdinger pair of operators on )l with parameter a ~ 0 satisfying O_DEs (37) and o eiVSDe-iUSds e s or eiVSDe-iVSds e s (38) / oo /5 where D is the operator related to {U, V} in (1). If g,,(U) = )t then there exists an auxiliary Hilbert space 1, a unitary operator W from g onto )~ = L2(IR, P) and a bounded strongly measurable s , ran (D))-valucd function or(.) on ~ such that

WUW -1 = 0 and WVW-a = fz

where (Lrf)(x) = mf(ac), f E )l (39) and . d a(z)*f ~(s)f(s)ds (Vf)(x) --- ia f(x) 4- ~ z, - (s q- io) (40) for f in

P(~") = {f E f( : f(.) is absolutely continuous and ft E )~}, (41)

where -4- in (40) are corresponding to the cases D >_ 0 and D <_ 0 respectively. The almost unperturbed SehrSdinger pair of operators {0, fr} in (39) and (40) is said to be the (integro-differential) model for {U, V}.

PROOF. We only consider the case that

0 < D E Ll(g) (42) and

D_ f~ E/Z()/) (43} co Let V(s) be the operator in (29), then

lim V(s)r/- Vr/- D_r/ for )7 E P(V),

Therefore a - lim(V(a) - IM)(V - D_ - t~l) -I -- I, ~)--=)-- CO 254 Xia for # ~\IR. On the other-hand, by (27) and the Laplace transform, it is easy to prove that ~o - .-~lim ~o(v(~) - ~,I) -1 = (~_ (v; u) - ~I) -~ for # ~\~. Hence

(w_(V; U) -/.tI) -1 = w - lim (V(s) - ttI)-1(V(s) - IH)(V - D_ - i~I) -1

= (v - D_ - uI). -1

Thus V = w_ (V, U) + D_. (44)

From the proof of Corollary 2 and by the Von Neumann theorem, it is easy to see that U has uniform multiplicity n- and there is a unitary operator W from )/onto the Hilbert space H = L2([R,/)) where/:) is an auxiliary Hilbert space with dim P -- n- such that WUW -1 = 0 where U is the operator in (39) and

(Ww_(V; U)W-I f)(z) = iad f(x) for f P(Ww_(V;U)W -1) which is the set in (41). Denote ~: = WVW -I. By (44), in order to prove (40), we only have to prove that there is a bounded strongly measurable/~(P --} ran(D))-valued function a(.) such that

(WD_W_lf)(x) _-. a(x)* f a(s)f(s)ds (45) 2~ri z - (s + io)"

Let (ei : j e J} be an orthonormal basis for the subspace ran (D). It is obvious that there is an orthogonal set {aj(.) e )~ :j J) such that

(Df)(z) = ~ l(f(.),aj(-))aj(x) J

and Ei HajH 2 = 2~rtr (D) < +c~. Then

(D_y,/) = ~x ~ f:oo I ~(f(~)"~i(~))~':~'d~l~d'- Xia 255

Thus the Fourier transform of the complex function x ~-~ (f(x),%(x))D on IR is square integrable on (-co,0] and (cf. [15]).

(D_f)(m) = E ai(x) x - (~ + io) i On the other-hand, from (43) we have

t eiUtD_e -jUt -- [ eiUSDe-iUsds. J- oo

Thus eiVtD_e -ivt is a monotone function of t and is bounded by IID_ {1I. Therefore there exists

8 - lim eiUtD_e -ivt = eiV~De-iV"ds E ff.()Q. t---*oo //oo It is easy to calculate(cf. 1151)that

$---*oo Y and s - lira eiVtD_eiVt is a bounded multiplicative operator, since it belongs to '$---~co s162 and commutes e ivt for t E IR. Therefore

II~ ~(:~)(c,,~(=))Dlid < liD-II fillip. (47)

For x E IR, define an operator a(x) from/) to ran (D} as follows ~(~)~ = ~(~, ~(~))Dcj Y

By (47), it is easy to see that

II~,(~)cll~ = ()--~ ~j(~)(c, ~j(~))D, ~)D < liD-II II~IIB.

Therefore a(x) E s , ran(D)) for x E IR and a(.) is bounded,

I1~(,)11 < IID-II 1/~.

It is obvious that the function x ~ a(x) is strongly measurable and (46) implies (45). Theorem is proved. 256 Xia

It is easy to see that under the condition of theorem 2, we have

(W(w+(V; U) - w_(V; U))W-lf)(m) = a*(x)a(x)f(x).

5. Cyclic cocycles on Weyl group, principal distributions and trace formula Let ~ be the Weyl group of dimension two with parameter a E IR, a r 0, i.e. the general form of the elements in ~ is

eik g(s,t), s,t,k e ~, and the group operation is

g(,~,t,)9(,,.,t,.) = e-'~ + ,~,t~ + t~).

Let {U, V} be a pair of almost unperturbed SchrSdinger operators with pa- rameter a. By Corollary 1, the family of operators {eikeisUeitV : s,t,k E IR} is an Ualmost" Weyl group of operators in the sense of [16]. Similar to that in [16], define a function r -) on ~ x ~ in the following

r ~k~g(~,t~)) = (48) tr([ei',v e",v e~k, ,e~,2v eit2v eik2]_ei(,,+,2)v ei~,+~)v eick,+k2)(e-i~,~,_ e-~,, r

By Theorem 1 of [16], r -) is a cyclic cocycle on the Weyl group ~ and there is a Baire function r on ~P such that

r = ~(g~9~) - ~(g~gl), for g~,92 e ~. (49)

From (48) it is easy to see that

~( eik' gl, e ik2 g2) = e i(k' -t-k2) ~(gl, g2).

Therefore, from the proof of Theorem 1 in [16], we can see that the function z can he chosen such that ~(~,kg(~,t)) = ~%(g(,,t)). (5o) Xia 257

Now, define

F(A,#) = tr ((U - AI)-I(V - #I)-aD(V - #I)-I(U - AI) -1) (51) for A, # E r where D is the operator in (1). In w we will prove that

F(,~, #) = tr ((V - #I)-a(V - )d)-lD(U - ,~I)-I(V - #I)-a). (52)

LEMMA 3. Let {U, V} be an almost unperturbed Schrfdinger pair of oper- ators with parameter a ~ O. Then the function r in (49) is a continuous function on IR • ~, r(eikg) = eikr(g) and

r(9(s,t)) = tr(eiW(lU, eitVl/ta- citY)) = tr(e i'v ei~V Dei(t-~)V dr)/ta fo ~ = tr(e.v([~v,Vl/sa- e~v')) = tr(e "v f e~VDe~("-~Vdo)/sa,(53) where [U, e i'v] and Iel*v, V] represent the bounded operators to which they extend. Besides, Ir(g(*,t))l < IIDIla/a. (54)

PROOF. For simplicity, denote r(g(s,t)') by r(s,t). By (49) and (50)

T(81 + 82,tl + t~)(e -~~ -e -~1'~) (55) = tr{[eiS, VeitaV ei*=Ue it2V] _ ei(ax+*2)Vei(tl+t=)V(e-ia~2h _ e-ia~l~2)}.

Let 81 = 8 +3', 82 = -'7, ta = t, t2 = 0 in (55), then

,-(8,t)(1 - e -~'') = tr(e"u(e~'%"(v-~"Oe -~^'v - e"v)}. (56)

From (18), it is easy to see that [U, e itv] - tae iW c f eir'CDe-irV drjVt (57) Tile left-hand side of (5'/) will be regarded as a bounded operator with s 1 norm less or equal to [t I IIDII~- Therefore (54) is a consequence of (53). On the otherhand, for g" E M, we have

. . -- __" ~(eV~V et(V a71)te ,v.~f --_ iei.lv (IU, eitV] I taeitV ) e-iT(U+atl) f. 258 Xia

Hence

eiU~tei(V-a~tl)te-iU~ -- e iVt = i/~ eiaV([U, e itV] - taeitV)e-iCrUe-ia~tdo. (58) do

By (56-58), we have

T(S~ t) Ct - ~-~::#) = i/~ t,'(e~"(.~'~([U, e "v] tae"v)~-~'~('+:+~)))d,.

= itr(([U, e itv] - taeiW)e i'u) fo r e-i="tda which proves one part of (53). The other part of (53) can be proved similarly. The continuity of r comes from (53).

LEMMA 4. Under the condition of Lemma 3, functions r and F satisfy

a ste-i'Xe-it~r(s, t)dsdt --- F(A, #), (59) do Jo where )~, # E ~\R.

PROOF. Without loss of generality we may assume that IA < 0 and )'# < 0. It is easy to verify that for f E M

f0~ ([u, e "v ] _ e"V ta)e-"~' r dt

= (i[u, (v - ~I) -I] + a(V - U~r)-2)~" = -(V - #!)-aD(V - #I)-'r

Besides, from (57), we have

[liu, e iv'] - tae+V+l[~ _< tllDIl~.

Hence, by (53) we can prove that

a t~-""~(s,t)dt = -tr((V - UI)-~D(V - UX)-l,"v), (60) which proves (59). Xia 259

Definition. Let {U,V} be an almost unperturbed Schr5dinger pair of op- erators with parameter a ~ 0, then the distribution

G(~, u) = ~- ff e-'('=+'~)~(g(8,t))ds~t (61) which is the Fourier transform of a r(g(., .)) is said to be the principal distribution associated with {U, V}. If a = 0 then G(-,-) is the Pinens principal function. The sign of G(., .) here is different from what is defined in [15]. From (53), it is easy to see that

~(o(8,t)) = ~(g(-8,-t)),

Therefore G(., .) is a real distribution. By (54), it is easy to prove that there exists a o-finite signed measure r on IR x IR such that

(c(., .), r .)) = [ r xR for every continuous function r with compact support. Thus (61) and (50) may be rewritten as -~(g(,,t)) = ~ 1 /fe~(~+'v)a(x, u)dzdy = elfe~("+'")dr(~,u) (62) and F(~'u)=-2-~1 ff (~-~)~(y-u)G(x,y)dxdy ~-- 2~1 ff (~-~)~(~-u)~"dF(x,y) (63)

Let f] be the space of Fourier transforms

r u) = 1 ff ~,(x,+~,,)~(8,t ) (64) i~• of complex measures w(., .) satisfying

ff(l+ 181)(1 + Itl)l~(~,t)l < +~. I~•

(cf. [2]). For a ~ 0, define a non-associative product in 12 as follows

Ja(r162 = 27ra[a'// I (r162 - r rl)r 260 Xia

i i" I"

- I"" Id (6s) where Cj E 12, j = 1, 2 are Fourier transforms of wi respectively. It is easy to see that 0(r162 lim J~(r162 = r e 12. --.o o(~,y) ' Therefore, for a = 0, define the non-associative product in 12 as 0(r162 J~162162 O(x,y) "

Let {U,V} he an ordered pair of self-adjoint operators. Define the Weyl- ordered operational calculus

r v) = 1 ff eiV,eiVtdca(s,t) for r E fl, where w is the measure satisfying (64).

THEOREM 3. Let {U,V} be an almost unperturbed SehrSdinger pair o[ operators with parameter a, then tr(i[C(U, V), r V)] - aJ~(r r V))

(66) 27r1 ff J.(r ~)(~, y)a(~, u)d~d~, where G(., .) is the principal distribution for {U, V}. In the case of a = 0, this theorem is well-known (cf. [2], [8], [6]).

PROOF. We only have to consider the case of a # 0. By (24), we have

Ill e~'' tre"' v, d.~tre.~V] _ e~(., +.~)v e~(~, +~}v (e-ia.~, _ e-i~., ~)lll

_< (min([s2[, Itll) + min(181[, It21)llDIl,. Therefore the following exchange of trace and integrals is adjusted, i.e.

itr( f ... / ([ei,,U eit,V,ei,,V eit, V] - ei(',+',)U ei(t,+t,)V (e -i~'~', -- e-ia"',)) Xia 261

d~Ol(81, t 1)do)2 (82, t2)

---- f"" f il"(g('l + 82,tl + t2))(e -ia82ta e-iasa t2 )d~l (81, t l )d~2(82, t2). Thus the left-hand side of (66) equals

_ (2~)3ai f.. f G(x, y)e i((sl+s2)=+(tl -i ' tl -- )dzdydwldw2

= 1//

by (62), which proves the theorem. For example, if {U, V} satisfies condition of theorem 2 and the function a(-) in (40) is the boundary value of an analytic function in H 2 on the upper half plane then G(z, y) = 0 for y < 0.

6. Determining function. Assume that {U, V} is a pair of self-adjoint operators satisfies the conditions of Lemma 1 with C -- -i(aI + D), a E R. Then it is easy to see that

(I + iD(V - #I)-'(V - AI)-I)(I - iD(V - AI)-'(V - #l)-') = I - aD(V - #I) -a (U - AI)-2(V - #i)-a. (67) and (I - iD(U - AI)-I (V - #1)-1)(1 + iD(V - #I)-a(U - A1) -1) = I - aD(U - AI)-'(V - #I)-u(U - M) -1 (68)

by (5). Therefore, if P E s ()/) then

tr (D(V - #I)-' (V - AI)-2(V - #I)-a) = tr (D(U - AI)-a (V - #I)-2(U - M)-'),

since tr(AB - I) =tr (BA - I). Thus (52) holds good. If D E s then denote

E(,\,#) = det(I - iD(U - AI)-'(V - #I)-'), A,# e ~,\IR. (69)

For any function f(A,#,-..) of complex variable's, the function f(A,~,...) is de- noted by if(A,#,...). It is easy to see that

E*(A,#) = det(I + iD(V - #I)-I(u - ~i)-1). 262 Xia

On the otherhand, it is easy to prove that

det(I - aD(V - lzI)-' (U - M)-2(V - .I)-') = detff - a(V - .I)-I (U - :a)-'D(U - - .I)-I).

Therefore (67) and (68) imply that

E(A,t~)E*(A,#) = get(/- a(V - I~I)-I(U - M)-xD(U - M)-I(V - ill) -1) ---- det(I - a(U - AI)-I(V - tzI)-I D(V - #I)-I(u - ,kI)-lIT0)

THEOREM 4. Let {U,V} be an almost unperturbed SehrSdinger pair of operators with parameter a E ~ on ~. If the operators D in (1) is of rank one and the only subspaee which reduces U and V and contains D~ is the space )t itself. Then the analytic function E(., .) defined in (69) is a complete unitary invariant. The function E(-, -) in this theorem is said to be the determining function of (v, v}.

PROOF. It is obvious that E(-, -) is a unitary invariant. Without loss of generality, we may assume that a :~ 0, D r 0 and D _> 0. Thus there is a non-zero vector e E D)/such that Df = (f, e)e, f E )I. (71) Therefore E()~,#) = l - itr(D(U - AI)-I(V - #1)-1). (72)

(70) and (71) imply that

E(A,#)E* (A,/~) --= 1 - aF(A,#). (73)

To show that E(-, , .) is a complete unitary invariant, first, we have to show that the inner product of the vectors in the subspace g, the closure of the linear span of

{(U -- ~I)-I(v - •1)-1e "- ,~,]s e [~\[~} [..J ((V - ~tI)-I(u - )~I)-le : ~,]A e (~\[~), is determined by the function E(-, .). (a) The function

((U - )~oI)-I(v -]AoI)-ie, (U - )li1)-i(v - ll, iI)-le), )~j,]~j ~ (r'_,\[[:( (74) Xia 263 is determined by E(-, .). To show this, first notice that by (71) the function in (74) equals tr((V --filI)-I(U--AII)-I)(U - AoI)-I(v -/ZoI)-lD). (75) We may assume that A1 # Ao, since the case of Aa = Ao can be determined by the limit process. Then the function of (75) equals

(tr ((V -~,xD -I (e-~l/) -I (v -uo/) -I ) --tr((V --~1 I) --I (e -- Aol)-I (V - IZo/)-'))

- (76)

LEMMA 5. Let {U, V} be an almost unperturbed Sehr~dinger pair of oper- ators with parameter a, and D be the operator in (1) satisfying rank D = 1. Then the functions

J(~,A;#) -- tr(D(U - ~1)-1(V - l,I)-l(U - AI)-l), ~, A,/, e (I:;\IR (77) and

K(A;~,v) = tr(D(V - #l)-1(V - AI)-I(V - vl)-1), A,/z,v e (I:;\IR (78) satisfy the differential equations

0 a~J(~, A;#) + i(~ - ~)J(~, A;#) + E(~,~u)E* (A,/z) - 1 = 0, (79) and a ~--~K(A;l~,v) - i(# - v)K(A;lz, v) + E* (A,#)E(A,v) - 1 = O. (80)

PROOF. Assume a # O. By the definition, we have

J(& A; U) = tr(D[(U- ~1) -1, (V -M)-I](U- AI) -1) + tr(D(V - #1)-1(U - ~I)-a(U - AI)-I). (81)

From (3) with C = -(al + D), it follows that

tr(D[(U - ~1)-', (V - M)-'](U - AI)-') = -iatr(D(U - ~I)-a(V - tH)-2(U - ~1)-1(U - AI) -1) - itr(D(U - ~I)-I(V - lzI)-ID(V - I,I)-I(U - ~I)-I(u _ AI)-I). (82) 264 Xia

By (51) and (U - ~I)-I(U - ,~i)- 1 __ ((U - ~I) -1 - (U - AI)-I)/(~ - A), we have,

tr(D(U- ~I)-I(v -#I)-2(U- ~I)-I(u- AI)-')

= (F(~,.) - ~J(~,~;.))/(~ - ~). for ~ # ~. (83)

By (71), we have tr(DADB) = tr(DA)tr(DB), for any A,B e/~(~(). Therefore

tr(D(V - ~I) -~ (U - ~I) -~ (U - ~I)-~)

-- itr(D(U - ~1)-1(V - #I)-I D(V -- ]~I)-I(u - ~I)-I(u - ~1)-1) = E(~,~)tr (D(V - ~0-i(u - ~0-'(U - ~)-1) = -iE(~,#)(E*(~,#) - Z*(A,#))/(~ - A), for ~ -~ A. (84)

From (73) and (81-84), it follows (79). Similarly, we can prove (80).

COROLLARY 8. Under the conditions of Lemma 5, the functions d(- ;., .) and K(. , .; .) are determined by the f, naio, E(., .).

:PROOF. It is easy to see that

[J(6 ~;~)1 _< IIDlll(I-r~ll~ll~AI) -1 and IK(A;~,v)l <_ IIDlll(I-r~ll-r~U@ -1 since [[(U- (i)-1[[ _< 1~till and [[(V -#i)-111 _

lira d((,2;#)=0 and lim K(A, #; v) = 0. (85)

Hence these differential equations (79) and (80) and conditions (85) determine the unique solutions J and K by E. Corollary 3 is proved. Thus the function (K(~l;gl,#o)- K(Ao;gl, #o)/(A1- Ao)in (76)is deter- mined by E(-, -). (b) The function

((V - #oI)-I(U- AoI)-le,(V - #II)-I(u- AII)-le), Aj,#j C ~\~t Xia 265

is determined by E(-,-). This case is similar to Ca). (c) The function

f(~,A;.,.) = ((u-A1)-'(v-.o-'e,(v--~O-'(v--&)-'e), ~,A,.,. c\~ (86) is determined byE(.,-). In fact,

f(~,A;]~,l~) = tr(D(V - ~/)-l(V -/~1)-1 (U - AI)-I(v -/2i)-1).

We have to prove that the function f satisfies the differential equation

a~f(~,A;p,~) + i(v --/~)f(~, A;/~, v) + iJ(~,A;I~)E(A,v)

+(E(~,v) - E(A,v))I(~ - A) = O. (87) From the proof of Lemma 1, we can conclude that if M = (V - AI)-~P(V) then [U,Y]f=-i(aI+D)f, for feM. If r E )4, then f = (U - AI)-I(V - vI)-lr M and hence

([U, V l+ (v- lz) (U- AI)) f = -i (hi + D) (U- AI)- I (V- vI)- 1r + (v- lz) (V - vI) - 1r

On the otherhand, we have

(U - AI)-I (V -/2I)-1r - (V - ~I)-I(u - AI)-Ir

= (V - ~I)-1(U - A0-~([U,V] + (~ - ,)(U - A]))~. Therefore (U - AI)-I(v - yi)-i _ (V - ~tI)-l(u - AI) -1 =

-ia(V -i~I) -1 (U-A/) -a (V-vI) -1 -i(V --/Z/) -1 (U-AI)-ID(U-AI)--1 (V _/,,I) - 1

+(. - ~)(v - ~z)-l(u - Az)-~ (v - .~)-~. (88) Multiplying both sides of (88) from left by D(U - ~I) -~ and then taking the trace of both sides, we obtain (87). Similarly to (85), we have

lim f(~,A;#,v) = 0. (89) IZ~l--.oo 266 Xia

As a solution of the differential equation (87) and the condition (89), the function f is determined by E(., .), since J(.,,.) is also determined by E(., .) (cf. Corollary 3). Thus the inner product of the vectors in )~ is determined by the function E(., .). It is evident that e E )l. The subspace )l reduces (V - #I) -1, # E C\IR. In fact, for v, A E ~\lR if v :fi A then

(v - ~a)-'(v - .1)-~(u - ~x)-l e

(~ - v)-!((V - IH)-!(U - AI)-!e - (V - vI)-I(U - AI)-le) e )l. We have to show that

(v - ~I)-~(v - ~O-'(v - ~I)-~e e #, A,~ e ~\~. (90)

Let r/be any vector in )/@ )/. For fixed #, v, and r/denote

f()O = ((V - I~I)-l(u - A])-l(v - vI)-le, zl), A E ~\ffR.

By (88), it is easy to see that

-af'(A) + (v-#)f(A) = 0.

On the other hand, lim f(A) = 0. Therefore f(A) - 0, which proves (90). I.rXl--,oo Hence, )l reduces (V-#I) -1, # E ~\IR. Similarly, )~ reduces (U-)~I) -1, A E tl:\lR. Thus )l = )/, which proves that the inner products of vectors in ); is determined by the function E(., .). It is well-known that the Pincus principal function of a pair of almost com- muting operators {U, V} on )/which satisfies the condition that the only subspace containing [U,V])/ and reducing U and V is g it self, is a complete unitary in- variant. For the pair of almost unperturbed Schr6dinger operators, the principal distribution is also a complete unitary invariant under certain condition.

LEMMA 6. Let {U~,Va}, lal <_ ~ be a family of almost unperturbed SchrJdinger pairs of operators with parameter a, Let Da be the operator D in (1), and E()%#;a), F(A,~t; a), J(~,A;~;a) and K(2;/x,v;a) be functions in (72), (51), (77) and (78) corresponding to the pair {Ua, Va}, respectively. If for non-negative Xia 267 integer n, there are functions Ek(A,#), Fk(,~,rl), Jk(s and Kk(s 0 _< k < n, which are regular for ~,.~,# E (13\IR satisfying Iim Eo(A,#) -- 1, )~,p--,oo M 1 IEk(.~,,~)l < II.~ll.r~,l' 1 < k < n, IEo(&,)l < M0 + I.r.),lli------i), M IFk(a,.)l _< (11~11~r.i)2 , 0 < k < n, M r-~(IJk(~,~;.)l, lKk(~;~,.)l) <_ iZ~ll~rall_r.i, 0 < k < ~, such that n E(,~,p.; a) = E akEk(~'P') + O(la["+l/[ t"~1[I"1)' (911 k=O n F(.~,#; a) = E akFk('~'#) + O(lal"+'/IZ'qZ"l), (92) k=O n J( ~, ~; #; a) = E akJk(~' '~; ,u) + O(lar'+l/l!SllZ,~llI.I), (93) k---O

K(.k;#,v; a) = E akKk(A; #' v) + ~ ' (94) k---O as a ~ O, and there is a constant c such that IID~tl, <- c. Then the function E,,,(-, .) is determined by the set of functions {Fk(-,-) :k = 0,1,---,m} for 0 < m <_ n.

PROOF. Let

H(~,)q#;a) = E(~,#;a)E*(~,#;a) - 1, (95)

R()L,#;a) = (E(~,~;a)E;(~,~) + E i (~,#,a)So()~,p))/2 (96) and I(2,#; a) ---- (E(~,#; a)E~(2,#) - E* (),, #; a)Eo(2, ~))12i. (97) From (91), (96), and (97), we have

n R(.~,#; a)= E akRk("~'#) + O(laln+l/IZall/~l) k=O and n I(:~,~,;a) = ~: akIk(~,U) + O(lal"+l/i.r.~ll.r,~l) k=O 268 Xia where Rk()~,#) = (Ek()~,#)E~()~,#) + Ei()~,#)Eo()~,#))/2 and ~(~, ~) = (E~(~, ~)E~ (A, ~) - E~ (~, U)Eo(~, ~))/~i. Therefore Rk(~,l.Z ) = Ri(A,#), Ik(,~,l~) = I[c(~,l~), and E~(X,p) --- (Rk(X,#) + ilk()~,Iz))Eo(X,#), since Eo(A,#)E~(A,#) -- I from (73), (91) and (92). It is easy to see that (91), (92), (93), and (94) still hood good if n is replaced by any positive integer m less than n. Therefore, from (73), we have

k+l F~(~,.) = - ~ E;(~,.) E*~+~_;(~,.1 O

(os) k=0 where Ho(~,~;g) ~ Eo(~,#)E~()~,#) - 1 and k nk(~,~;~) -- ~ EjCr o < k < ~. (99)

By (79), (93), and (98), it is easy to see that i(~ - ),)Jo(~, J~,#) + Ho(~,X;t~) = 0. and ~ Jk-~(~, A; ~) + i(r - A)Jk(~, A; ~) + Hk(~, X; ~) = 0, ~=1,2,...,.. (loo)

Therefore k ( i )l+lat k = 0,1,--.,n. tin0 Xia 269

By means of (99), it is easy to calculate that

J&()~,A;#I=E(I+II]c3#IEE;(A,#)~E}_t_i()~,# ). (101) l=O j=O From (r3), (92), (95) and (98), we have

Hk(A, A;#) -- -Fk-1 (A,/z), 0 < k _< n. (102)

In (100) and (102), let ~ = ~, then

~-~gk(A, ~;/z) ---- Fk(,,~,#), 0 < k < n.

Thus by (93), we have

t, Fk(A,v)dv, 0 < k < n. (103) Jk(A,A;~) = f(~.)ioo Taking k = 0 in (101) and (103), we get

i o lnEo(A, #) = /":r~)ico Fo(~, v)d~. (lO4)

Similarly, using the argument related to K(~; A, #), we may prove that

i s = .r~)iooFo(~,p)d~ (105)

Notice that aim Eo($,#) = 1. From (104) and (105), we have .~,p--*oo

), p.

Hence Eo(,~,#) is determined by Fo(-,-). For 0 < k < n, assume that Eo, El,-.. ,Ek-1 are determined by Fo,..-, Fk-1. We have to prove that Ek is determined by Fo,-..,Fk. We have already known that Rk is determined by Fo, F1,... ,Fk-1. Form (101) it is easy to see that

Jk(A, A;#) ---- i(E;(A,#)~ A E,(A,/z) + E; (A,#) ~-~AEo(A,#)) + terms related to Eo, E1,--',Ek-1 ~Ik(A,#) + terms related to Fo, F1,...,Fk-1 OA 270 Xia

By (103), we may conclude that

~--~ Ik()~'lt) = -- f(".r,)ioo Fk(A,v)dv+termsrelatedtoFo,F1, "" , Fk-1.

Using the formulas related to K(~; A, #), we get Ik()~,~) = fIa)i Fk((,#)d( + terms related to Fo,".,Fk-~. since lim,__,~ Ik(A, #) = 0. Therefore

= - Fk((,v)d(du + terms related to Fo,...,Fk_l. Ik ()~,/~) ..r.X)ioo 2".)/00

Thus Ik is also determined by Fo, F1,'" ,Fk which proves the lemma.

THEOREM 5. Let (U,V} be an almost unperturbed SchrSdinger pair of operators on )l. If the operator D in (1) is of rank one, the only subspace which reduces U and V and contains D~( is the space ~( it self, and the determining function E()~, #),)% # G ~\~ is either an analytic function Iz at # = co for fixed E C\~ or an analytic function of )~ at )~ = co for fixed # 6 ~3\~, then the principal distribution G(., .) of {U, V} is a complete unitary invariant.

PROOF. Without loss of generality, we may assume that the parameter of (U, V} is 1, and that the determining function E()~,/z) is an analytic function tz at # -- co for fixed ,~ e ~\IR. The pair {U, aV} is an almost unperturbed SchrSdinger pair of operators with parameter a and the determining function of {U, aV} is

E(A,/~; a) = 1 - iatr(D(U - ~I)--1 (aV -#I)-1). = E(A,#/a), a # O.

Therefore E(~,#; a) is an analytic function of a in the neighborhood of a = 0 for fixed ,~,/~ E ~\IR. It is easy to verify that those functions in lemma 6 corresponding to (U, aV) are

F(,~,#; a) = F()% I~/a)/a, J(~,~;/z;a) = J(~,~;p/a), Xia 271 and

= la)la), and they satisfy the conditions in lemma 6. Thus E(A,/~;a) is determined by F(A,I~/a) which is determined by the principal distribution G(.,-). Theorem 5 is proved. For example, if {U, V} satisfies the condition of theorem 2 and the support of the function ~(.) in (40) is compact, then the determining function E(A, #) is an analytic function of A at A = co for every fixed/z ~\IR.

7. Perturbation of group of unitary operators and a generalization of Von Neumann theorem. From the proof of the lemma 1 in [16], it is easy to see the following:

LEMMA 7. ff V(t) and Vl(t), t ~n are two continuous groups of unitary operators on )l satisfying

rank (V1(t) - V(t)) <_ 1, t Rn, then either Vl(t) = V(t) for all t ~n or there is a one-dimensional subspaee M of g which reduces V(t) and Vl(t) for t Rn and V(t)[M• = VI(t)[M• The proof of Lemma 7 will be contained in the proof of Lemma 8.

LEMMA 8. Let V(t) and Vl(t), t IR'~ be two continuous groups of unitary operators on ~l satisfying

rank (VI (t) - V(t)) < 2. (106)

Then either Vl(t) = V(t) for all t E Rn, or there is a subspace M c )I satisfying 0 < dimM < 2 such that M reduces V(t) and Vx(t), t ~n and Y(t)lM• = VI(t)IM•

PROOF. Denote Sj = {t e IRn : rank (V1(t) -V(t)) =j}, j = 0,1, 2. Let U(t) = V(t)-IV1(t), t e IRn. Then U(t) is a unitary operator. Denote

v(t) = ran (U(t) - O, then dimp(t) --- 3" for t E 8j. There is a normal operator A(t) on p(t) such that 0 r o(A(t)) and U(t) = I + A(t)P(t), (107) 272 Xia where P(t) is the orthogonal projection to p(t). It is easy to verify that

U(to)U(tl) -1 = V(tl)-lU(to - tl)V(tl) (108) and U(-t)* = V(t)U(t)V(t) -1. (lo9) From (108) and (109), we have

V(tl)-lp(t2)V(tl) <_ P(tl) V P(tl + t2) (110) and P(-t) -- V(t)P(t)V(-t) (lll) respectively. First, assume $2 = r This means that the condition of lemma 7 is satisfied. We only have to proof Lemma 7. For to, tl E $1, we have to prove that

P(to) = P(tz).

The identity (108) implies that

(I + A(to)P(to))(l + A(Q)P(Q)) -- I + A(to - Q)af(to - tl), (112) where A(ti) e C,A(tj) :~ 0, dimp(tj) = 1, i = 0,1 and

af(to -- Q) = Y(tl)-lp(to - tl)Y(tl).

If P(to) r P(tl), then there is a non-zero vector

x~ ran (P(to) v P(tl) - af(to - tl)).

Then, by (112), we have

A(to)P(to)x = A(tl)P(tl)z.

But P(to)x ~ 0 and P(tl)x ~ O, this is a contradiction. Thus there is a one dimensional space p such that

p(t)=p for tES1, if $1 ~ r Therefore p reduces V(t) and V1 (t) which proves Lemma 7. Now, assume 8'2 ~ r We have to consider the following three cases. Xia 273

I. Suppose there are tl, t2 and t3 E 5'2 such that P(tl) A P(t2), P(t2) A P(t3) and P(t3) A P(tl) are three different rank one projections. Let Q = P(tl) V P(t2) V P(t3). then rank Q = 3 and

Q = (P(Q) A P(t2)) V (P(tz) A P(ts)) V (P(t3) ^ P(tl)).

We have to prove that

P(t) <_ Q for t E [Rn. (113)

We may assume that t E 81 O Sz. From identity (108), we have

(1 + A(tj)P(tj))(I + A(t)*P(t)) = I + A(tj - t)P(tj - t),j = 1, 2, 3 (114)

where ),(tj -t) = V(-t)A(tj -t)V(t), and/3(tj -t) = V(t)-lP(ti -t)V(t). We have to prove that rank (P(ti) A P(t)) > 0, (115) j = 1, 2, 3. If (115) does not hold, then rank (P(tj) v P(t)) :> 3 and there is a non-zero vector x e ran (P(tj) v P(t) - P(tj - t)). From (114), we have ~(t~)*P(t~)z = )~(t)*P(t)z. Thus, )~(t)*P(t)z E ran (P(tj) ^P(t)). This implies x = 0. That is a contradiction. Thus (115) holds good. It is obvious that (115) implies (113). From (I10), we have

V(t)-lP(tj)V(t) < P(t) V P(t i + t) <_ Q

for t E IRn and j = 1, 2, 3. Therefore

V(t)-IQV(t) <_ Q, for t E IR"

which proves that ran(Q) reduces V(t), t E IR'L It is obvious that dim( ran (Q)) = 3, ran (Q) reduces U(t), t E IRn and hence ran (Q) reduces Vl(t), t E IR". Therefore there are x(t) e ran (Q), I[x(t)l[ = 1 such that V(t)x(t) = Vl(t)x(t). Then we can prove that there is a two dimensional subspace of ran (Q) which reduces Y(.) and VI(.). 274 Xia

2. Assume that there is a rank one projection Q such that Q < P(t), for t E $2 and _P(Q) # P(t2) for some Q,t2 E $2. For every t E 81 and any ts e S~., from (114) we can conclude that (115) holds good for j = 3. Therefore

P(t0) ___ P(,), for ~ e s~, to e s,, wMch prove that q _< P(t) t'o~ t r s~ u s2. (116) By (i~6) a~d (m) we have

Q < v(t)P(t)v(t) -1 for t E S1U S2, since t E S i implies -t E Sj. Therefore

V(t)-'QV(t) < P(t), for tES1US2.

Especially, we have

v(t)-~QV(t) = q fo~ t e s~. (117)

Let = {t e s~ : v(t)-'QV(~) # Q} We have to prove tha~ for to, t E S~ either

V(to)-lQV(to) = V(t)-'QV(t), (llS)

P(t) = P(to), or V(to)-' QV(to)lV(*)-lQV(t). (11o) Let t/be a unit vector in ran (Q). Then for t E S, vectors t/and V(-t)7/span p(t). Therefore there is an invertible 2 x 2 matrix cij (t) such

2 v(t). = + ~ .,j(t)(.,e,).j, for x e ~ (120) id'=l where el = r/and e2 =- V(-t)rl. Since (118) is equivalent to t - to ~ S. We only have to prove (119) under the conditions that P(t) # P(tc) and to, t and ~ - to E ,~. Denote

r = (v(t),.,). Xia 275

By (120), 2 U(t)* x = x + ~ eq(tl(x, eh)ei , for x X, i,3'----1 and U(t)U(to)* - V(-to)U(t - to)V(to) = 0, we can prove that

bq(x, m)ny = o where ,11 = ~, ~ = v(-to),7, ,7~ = v(-t),7, and bll = ell(to) + ezl(t)(1 + ell(tO)) +e21 (t0)r + e21 (t)(ell(t0)r + C21 (t0)r -- to)), b21 = 2era(to) + em(to)r bsl = ezl(t)(1 + cm(to)r + c22(to)r - to)), b12 -- e2z (to),

b~ = c~(to) - ell (t - to), b33 = eZl (t - to),

b13 = el2(t)(l+elZ (to)-[-c21 (to)r )+e22(t) (ell (to)r162 - t) ), b23 -'- em(t)(cm(to) + em(to)r ) - el2(t - to), b33 = e~2(t)(1 + cm(to)r + el2(to)r - t) ) - e22(t - to). These bq = 0, if P(to) ~ P(t). By calculation we may prove that

r = r = r - to) = o.

Thus {r/,Y(-to)~/, Y(-t)~l} is an orthonormal set and (119) holds. There exists a positive number e such that

P(t) = P(to) for to, t e S n S(0, e) (121) where S(0, e) is the sphere in IRa centered at 0 with radius e. Otherwise there is a sequence ta ---+ 0, ta ~ 0 such that ta E S, but P(ta) ~ P(ta') for n r n'. Therefore V(-ta)rl-LV(-tw)rl for ta r t~ Thus IIw(ta, - ta)~ - 711 = v~ 276 Xia

This contradicts the continuity of V(.). Hence (121) holds. The set {t: V (t)-I QV (t) = Q, t E S(0,~)} is closed. If S [3 S(0,r is not empty, then there is an sphere S(c, 6) c S(O, e) n S. Thus there is a projection Q1 with rank 2 satisfying

Q1 = P(t) = P(t') = P(-t), for t,t' e S(c,6)

By (110), it is easy to prove that

P(t) = V(t)-lQiV(t) = Q1 for Itl < ,V2.

Thus ran^(Q1)reduces V(-)and 111(.). If S v1S(0, e) is empty, then V(t)-IQV(t) = Q for Itl < ~. From Vl(t) = V(t)U(t), it is easy to see that

Vl(t)-lQVl(t) = Q for Itl < ,.

Thus ran (Q) reduces V(-) and VI(-). 3. There is a rank two projection Q such that P(t) = Q for t E $2. Then it is easy to prove that ran (Q) reduces V(.) and VI(.). Lemma 8 is proved.

THEOREM 6. Let {U(s) : s e IR'~} and {V(t) : t e [Rn} be continuous representations of the additive Euclidean group [Rn in the Hilbert space ~t. Suppose there is a positive a such that

rank (U(s)V(t)U(-s)V(-t)e -ia*t - I) <_ m, /or s,t e R ~,

If m = 1 or ~ then there ezists an orthogonal d ecomposion of

ot such that )la reduces U(.) and V(.), and either dim X~ <_ m or

u(s)lu v(t)lx~u(-s)lu V(-t)lu e -i='t -- I for 8, t E R n.

PROOF. By a theorem in [16], we only have to consider the case of m = 2. Without loss of generality, we may assume that {U(s), V(s) : s E IRn} is irreducible on g and there are sl,Q E IR" such that

rank (U(sl)V(tl)U(-sl)V(-tl)e -i~'t' - I) = 2 Xia 277

Let Va(t) = U(s)V(t)U(-s)e -ia*t, then

rank (V(t)-lV,(t)- 1) < rank (V(tl)-lvsx(tl)- I)= 2.

By Lemma 8, there is a subspaee M,~ with dimension 2 which reduces V(t),t E ~n and Va~ (t), t E IRn such that

v(t)]u = y,1 (t)

As a matter of fact, M~, - rang (V(tl)-lvs, (tl) - I) and V(t)M~, = M~. On the otherhand, let Utl (s) -- V(tl)V(s)V(-tl)e iaat~ , then

rank (V(8)-lvf, (8)- 1) ___ rank (V(-81)-lvt,(-81)- I)-u-2.

Then there is a subspace N h with dimension 2 which reduces U(s),s E IRn and Ut, (*),* E [Rn such that v0)lN,, = v,,(8)IN,,. Thus, we also have

Nt~ = ran (U(-si)-XUt~ (-al) - I) and U(s)Nt, = Nt,.

However

ran (U(-81)-lutl (--81)--1") = ran (V(Q)(V(t,)-IV,, (tl)- I)V(-tl)-l). therefore Nt~ = M, 1 and it reduces U(s), V(a), s E [R'~, which proves )/ = Ms,. That means dim g = 2. Theorem is proved. The author wishes to express his appreciation to professor Louis de Branges for his invitation to visit Purdue University 1987 and his hospitilities. The author is also grateful for the helpful discussions he had with professor de Branges. Part of results in this paper reported at the Conference of Nonlinear and Convex Analysis in honor of Professor Ky Fan in Santa Barbara June 23-26, 1985. The author is also grateful for the helpful discussions on the unbounded hyponormal operators he had with Professors Paul Muhly and Raul Curto in 1982- 84.

References

[1] Carey, R. W., Pincus, J. D., Commutators, symbols and determining func- tions, J. Functional Analysis 19 (1975), 50-80. 278 Xia

2] Carey, R. W., Pincus, J. D., Mosaics, principal functions and mean motion in yon Neumann algebras, Acta Math. 138(1977), 153-218.

3] Carey, R. W., Pincus, J. D., Almost commuting pairs of unitary operators and flat currents, Integ. Eq. Oper. Theory 4 (1981), 45-122.

4] Claneey, K. F., Seminormal operators, Lectures Notes in Math. No 742 (1980) Springer-Verlag.

5] De Branges, L., Perturbation of self-adjoint transformations, Amer. J. Math. 84(1962), 543-580.

6] Helton, J. W., Howe, R., Integral operators, commutator traces, index and homology, Lectures Notes in Math. No. 345, (1973), 141-209, Springer-Verlag.

7] Kato, T., Perturbation Theroy for Linear Operators, (1968), Springer-Verlag.

[8] Pincus, J. D., Commutators and systems of singular integral equations, I, Acta Math. 121 (1968), 219-249.

[9] Pincus, J. D. On the trace of commutators in the algebra of operators gener- ated by an operator with trace class self-commutator, Stony Brook (Preprint).

[10] Putnam, C. R., Commutation Properties of Hilbert Space Operators, Eng. Math. Greng. No. 36, (1967), Springer-Verlag.

Ill] Reed, M., Simon B., Methods of Modern Mathematical Physics III. Scatering Theory, (1979) Acad. Press.

I12] Rosenblum, M., Perturbations of continuous spectrum and unitary equiva- lence, Pacific J. Math. 7 (1957), 997-1010.

I13] Von Neumann, J., Die Eindeutigkeit der SchrSdingershen Operatoren. Math. Ann. 104 (1928), 570-578.

[ 14 ] Xia, D., Measure and Integration Theory on Infinite-dimensional Spaces. (1972) Acad. Press.

[ 15 ] Xia, D., Spectral Theory of Hyponormal Operators, (1983) Birkh~user Verlag. Xia 279

[16] Xia, D., Trace formula for almost Lie group of operators and cyclic one- cocycles, Integ. Eq. Oper. Theory 9 (1986) 570-587.

Department of Mathematics Vanderbilt University Nashville, TN 37235 U.S.A.

Submitted: June I, 1988 Integral Equations 0378-620X/89/020280-2051.50+0.20/0 and Operator Theory (c) 1989 Birkh~user Verlag, Basel Vol. 12 (1989)

Toeplitz Operators and Hankel Operators

Dechao Zheng

We characterize those symbols f in L'~(D) for which the associated Hankel operators or Toeplitz operators Hf or Tf are compact on the Bergman space La2(D). These results have extensions to the case of unit ball B n and polydisc D n in C n. We show, in addition, for f in C(~), the maximal ideal space of H~176 that Hf is compact if and only if f is in AOP i.e f is analytic on the parts, and we show that both Hf and Hy are compact if and only if f is constant on the parts i.e f is in COP. Furthermore oe(Tf) = f(11~\D) and thus the essential spectrum is connected when f is in AOP.

Introduction. Let D be the open unit disc in the complex plane C. The Bergman space La2(D) is the subspace of L2(D,dA), which consists of analytic functions in D, where dA is normalized two-dimensional Lebesgue measure on D. For any function f in L=(D,dA), the Hankel operator Hf: La2(D) -) [La2(D)] -L and the

Toeplitz operator Tf: La2(D ) La2(D ) are defined by

Tf(g)=P(fg) and Hf(g)=(I-P)(fg) for g in La2(D) where P is the Bergman orthogonal projection from L2(D,dA) to La2(D).

Let % denote the ideal of compact operators of L(H), the algebra of all bounded operators on the Hilbert space H. For a T in L(H), the essential norm IITlle is

defined by IITII = inf II T - KII e K~'K

In this paper we find uper and lower bounds of essential norms of a Toeplitz or Zheng 281

Hankel operator. An analysis of these quantities allows us to characterize the functions f in L~176 dA) such that Tf or Hf is compact.

Section 1 contains some relevent definitions and reviews Hoffman's results on the maximal ideal space of H ~176In Section 2 we get uper and lower bounds for the essential norms of Tf and Hf. Consequently in Section 3 we can characterize the functions f in L~176 dA) such that Tf or Hf is compact, and we also show that the main result in [15]; namely, for f in L~176 dA), both Hf and H 7 are compact if and only if f is in VMOo~(D), is a corollary of Theorem 3. Section 4 shows that forf in C(~), the continuous functions on the maximal ideal space 'In of H ~176Hf is compact iff f is in AOP and Hf and H T are compact iff f is in COP. Moreover in Section 4 we prove that forf in

AOP, the essential spectrum of Tf is equal to f(~\D) and so is connected; and we reprove in a different way the result in [12] that forf in C(I~), Tf is compact iff f is continuously extendable to D with flo~D=0. In Section 5 we generalize the results in

Section 4 and 3 to unit ball B n and polydisc D n in C n.

I am grateful to Professor Joel Pincus for many helpful conversations.

Section 1. Maximal ideal space of H ~176

Let ~ be the maximal ideal space of H ~176which is defined to be the set of multiplicative linear maps from H ~176onto the field of complex numbers. If we think of as a subset of the dual of H ~~ with the weak-star topology, then 'In becomes a compact Hausdroff space. Using the Gelfand transform we can think of H ~176as a subset of C('I~), the algebra of continuous functions on ~.

For m, x in ~, the pseudohyperbolic distance between m and x, denoted by p(m,x) is defined by

p(m,'c) =sup {Im(f)l: fin H 0~ Ilfll < 1 and x(f) =0}. 282 Zheng

If w and z are in D contained in ~11,then the Schwartz Lemrna can be used to show that p w,z) =1 1.

For m in '11~,the Gleason part P(m) of m is defined by P(m) = {~ in ~ : p(m,x) < 1 }.

For z in D the Mobius transformation ~)z:D ->D is defined by

W -Z 0z (w) = 1 - ~ w

For m in ~11.\D, Kenneth Hoffman constructed a canonical map L m of the disc D onto the Gleason part P(m), which is defined by taking a net {z~ in D such that za+ m and defining L(z) = tim f,% (z) for z in D and f in H ~176The above limit exists and is independent of the net {z(x }, provided Zc~+m. For f in H ~176the composition f.L m is in H ~176In fact Hoffman also proved the following results which will be used later on.

HI. Let m be any point of ~\D. If a net {zcc } in D converges to m, the corresponding map net {0zc(} converges pointwise to L m.

1-12. C(~) is the sup norm closure of the algebra generated by H ~176and the complex conjugate of H ~176and also is the supnorm closure of the algebra generated by the bounded harmonic functions. Since for any z in D the pointwise evaluation of functions in La2(D) at z is a bounded functional, then there is a function Kz in La2(D) such that

f(z) -- < f, Kz> for all f in La2(D) which K z is called the Bergman reproducing kernel and is of form Zheng 283

1 K (w)= Z ( 1 -zw) 2

We use 1~z to denote the normalized Bergman kernel Kz(w)/Kz(z ). It is easy to prove the following transformation law

Czo0 ((~z(W) (~z(~.) (~z(W)) = K x (w) (1) and ~z(;~)--.afa~[(~z(~) 1. (2) For any f in L=(D), the Berezin symbol of f is defined by ~(z) =If(w) I'hz (w)l 2 dA(w). D

VMO~)(D) in [15] is defined by

{fELl(D): I IT(z) - f*r dA(u) -~ 0 as Izl-> l } D Section 2. Essential norms of Toeplitz and Hankel operators In the section we will find the upper or lower bounds of essential norm of a Toeplitz or Hankel operator. Theorems 1 and 2 are the main results in the section. Theorem 1. For f in LC~ we have

]i--~ IIHf ~zll _< II Hfll e _< ~ IIP(f=Cpz) - f~ II~llfll C Izl -) 1 Izl->1 for some constant C >0. Theorem 2. For f in L~176 we have li---~ IITf 'hzll < II Ttll e _<1-~ IIP(f.(~z)ll ~Y211fll C Izl -> l Izl-) 1 for some constant C>0. In order to prove Theorems 1 and 2, we state the following lemma which was proved in [2] and will be used in the proof of Lemma 2 and Lemma 3. Lemma 1. For s < 4/3, let 284 Zheng

C s = SupzeD I 1-zO~ Is (1-10~ 12)s/2

Then C s < +oo.

Lemma 2. Suppose that F(z,w) is a non-negative function in Lt(DxD,dA) where 1/t + 1/s = 1 and l

c1/'S [ j'lF(z, w)I' 11~+(w)? ~(w) ]l/t (1 - Izl 2 )1/'2, D for all z in D.

Proof. For a fixed z in D, we make the change of variables ~, = (I)z(W) to get f _V(z_,w). dA(w) D II-zw12(1-1w12) 1/2

1 r F(z, ~ z ( ~, ) ) _< (1- Iz~) 1/2 I 1-zXl(1 - 17LI2)1/2 dA( ~, ). by Holder's inequality and choosing s such that 1/t+l/s=land s<4/3 1 f lit [ 1 1/s < [ F(z, ~z(~ ))t dA(X,) ] [ ~. j~) ,/2 ~(;~)] (1- Iz12)1/2 D Dll'~'lS(1-1 by Lemma 1 cl/Ss ] l/t ( 1 I;I2 )1/2 [ ~ F(z, *z(~L))tdA( ~L) . - D

The proof is complete. Lemma 3. Let f be in L=(D). Then Zheng 285

I 1 - "@ z12( 1 - Izl2) 1/2 dA(z)

2C~1~1 (3) ( 1 - Izl2 )1/2 and IP(f*r z )*r dA(z) D I i - w zi2( 1 - Izl2)I/2

2 C 1 Ilfl[ _< (4) ( 1 - Izl2 )1~

Proof. It suffices to prove (4) in case iIfLl~. -< 1. First in order to simplify

P(r162162 we change variables with help of the transformation law (1) and (2) to get P(f*r162

I flu)( 1 - ~w )2 dA(u). D(1-~w)2(1-zu) 2

Therefore IP(for z )OCz(W)l dA(z) DI I 1 - ~ z12( 1 - Iz12)1/2

< f f 12 If(u)l 12) 1/2 dA(u) dA(z) D Dll--U w II-z ul2(1-1z

To substitute z by Cu (t) first and then u by Cw(S), we have 286 Zheng

f ip(f,Oz )*Oz(W)l dA(z D I 1 - w z12( 1 - Izl2) 1/2

<1 !!. dA(t) dA(s) ( 1 - Iw12) 1/2 I 1 - ~ w I I 1 - s T I ( 1 - I s 12 )1/2(1 - I t 12 )1/2 2 C 1 _< ( 1 - Iwl 2 )1/2 to complete the proof of Lemma 3. Lemma 4. Let Uz be an operator from L2(D) to L2(D) defined by

Uzg(w ) = g o~z(W) &z(W) for g in L2(D) Then (a) Uz is unitary,

(b) PU z = UzP. The proof of Lemma 4 is easy, so it is omitted, but see for example [15]. Before going on the proof of Theorem 1 and 2, we need a classical result of Schur ( [9] ), which is useful for studying Toeplitz operators on the Bergman space ([2],[14], [15] ).

Schur's Estimate. If F(z,w) is a non-negative function, if p and q are strictly positive measurable functions on D and if a and b are positive numbers such that

~F( z, w ) q( w ) dA(w) < a p(z) for almost every z in D

and

I F( z, w ) p( z ) dA(z) _< bq(w) for almost every z in D

then the operator F: L 2 + L 2 defined by

(Fh)(z) = [ F( z, w ) h( w ) dA(w)

is bounded and IIFII 2 _< ab (5). Now we turn to the proof of Theorem 1 and Theorem 2. Zheng 287

Proof of Theorem I and Theorem 2.

Without loss generality we may assume that Ilfll~_ 0, there is a compact operator K such that

[{Hf- K II -< [IHflle + e. (6)

Since ~z converges weakly to zero then ]]Klazl]2 + 0 as ]z I -)1. Thus (6) gives

II Hf IIe + e > li---~- II Hf ~z II Izlr

SO 11Hf I1e > i---~l II Hf laz II2 (7). lzl-->1

Similarly we can also prove that II Tf IIe -> i--~l II Tf ~z 112" (8) Iz~ I

Therefore (7) and (8) say that it is sufficent to prove that there is a constant C > 0 such

that II Hf IIe _< C li~ II V(f*~z ) - fO(~zll~llflly 2 Iz~l

and II Tf IIe _< C ~ II P(foCz)II~llflly2. Izl+l For z in D and h in ( La2)J- then

Hi*h(z ) = < h, HfK z >

=I h(w)( 1 - z w )-2 [( I-P)(f,r

and for g in La2

Tf*g(z) = < g, TfK z >

=[ g(w)( 1- z w )-2 [~-~( f*~z )] *r (w)dA(w).

Let %E denote the characteristic function of Ec D and let Mf be the multiple operator 288 Zheng from L2(D) to L2(D) defined by Mf(g) = fg. Then M%r D Hf* and M%rDTf* are compact since the kernel functions of these two integral operators are in L2(DxD).

Now we want to estimate the norms of Hf* - M%rDHf* and Tf* - M%rDTf*. Let p(z) = (1 - Izl2) 1/2 and q(w) = ( 1 - Iwl2) 1/2- Combining Schur's result with Lemmas 2 and 3 gives It Hf-M% H: I12<2C~1 C s1/s Sup [ j'l fO~z -P(fo~z ) t5 dA ]I/:}lfll rD lzl>_r D and

II Tf - M%r D Tf 112 -< C21 C1/ss SUPlzl>_r [ I P( f.~z ) 15 dA ]1/511fll D for s =5/4. Thus there is a constant C> 0 such that

IIHf IIe = IIHf IIe < limr~lllHf -M%rDH f II

II Tf lie = IITf IIc < limrr 1 IITf- M%,D T f II

(3.a) Hf is compact,

(3.b) lira II H f 1t z II2 = 0, Izl->l (3.c) lim f I f~162- P(f~ ) Ip = 0 for all p >- 1, Iz~l DJ (3.d) The semi-self-commutator Tiff2 - T T Tf is compact.

Theorem 4. For f in L~(D), the following statements are equivalent: (4.a) Tf is compact, (4.b) lira I] T f ~z 112 = 0, Izl-~l (4.c) lira f I P(f,r ) Ip = 0 for all p > 1. Izl-Y1 ~)

Proof of Theorem 3 and Theorem 4 The equivalence of (3.a) to (3.c) follows from the well-known fact that

# -Tf T =H T

Without loss of generality we may assume that Ilflloo <1. Since for any p, q >_1 and for any measurable function h, we have

I IhlqdA<( I IhIPdA)I/P( I IhI(q-1)P/(P-1))P-1/P then it follows from the boundedness of the Bergman projection from LP(D) to LaP(D) for p>l. there is a constant C>0 such that

llP(foCz) - f~162 <_C IIP(fo~z) - fOCzllpl/q and

llP(f.%)ll q _

tzl -) 1 Izl-) 1

~q-~ IIP(f.r ) -f*r176 Iz~l =1-~ IIHf'hzl112/l~ Izl--)1 and

,i~ lITr "hzll _< II Tflle _<~ llP(f.(~z)ll~t211fll~2C Izl § 1 Izl§ 1 _<~ llP(fo(~z)ll~/1011flli/2 C fzl-) 1

C fzb I Thus this impliesthat (3.a) and (4.a) are respectivelyequivalent to (3.b) and (4.b). The proof is complete Remark. From Theorems 3 and 4 we see that the reproducing kernels alone determine the compactness of Hankel operators and Toeplitz operators. In [15] Kehe Zhu proved the folPowing theorem which is a corollary of Theorem3. Theorem 5. For f in L~(D), then both Hf and H? are compact if and only if f is in VMOa(D).

Before proving Theorem 5 we need the following propsitions in [15].

Proposition 1. The Bergman projecton operator P: L'~(D) -) La2(D) is compact. Proposition 2. If {fn} is a sequence of real-valued functions in L2(D) such that

Iffn - h J12 § 0 as n goes to oo for some h in La2(D ), then h is a constant.

Proof of Theorem 5. Suppose that both Hf and H~ are compact, Without loss of generality we may assume that f is real-valued, if f is not in VMOa(D), i.e.

Hm Ill (z) - f*r 112 > e > 0, Izl-,'.1 Zheng 291 then there is a sequence {Zn} in D such that

]1 f(zn) - foez.ll -> E (9).

since { f~ } is a bounded sequence in L'~(D), it follows from Proposition1 that there is a subsequence {Znk} of {Zn} so that

II P( f~ )" h II "> 0 (10) as k goes to ,~ for some h in La2(D), On the other hand, Theorem 3 implies that

II f~ P( f~ ) 112 + o as k goes to oo. So (10) implies that

II f~ h 112 -> o as k goes to ,~. Proposition 2 says that h is a constant. Then

II f(znk) - f'~zokll -< II ~(znk) - hll + II h- foeznkll -> o. Since

f(znk) = I f~ dA -> h which contradicts to (9), therefore f is in VMO~(D).

Suppose that f is in VMO~(D). Then

lim II f(z) - fO0z II2 = 0 Izl->l Thus

II f~ P(f~ II -< II f~ f(z)II + II ~(z)- P( foez)ll

<-211f~ f(z)ll2-> o since P is a bounded operator from L 2 to La 2 with norm one. So Theorem 3 says that

Hf is compact. Similarly we can prove that H T is also compact. This completes the proof of Theorem 5. 292 Zheng

To end the section we define AQ = { f in L~(D) :ll f~ P(f~ 112 "> 0 as Izl ->1}.

Theorem 3 says that AQ = { f in L~176 : Hf is compact } and QA is a subalgebra of L=(D). Section 4. C('ffl.), AOP and COP In the section we will characterize more precisely those functions f in C('TIt) such that Tf or Hf is compact. The maps L m play a very important role.

First we introduce two subalgebras of C(~) which were defined in [4]. AOP, which stands for" analytic on parts", is defined by

AOP={ f in C(m) I foLm in H~ for all m in ~\D}, COP, which stands for" constant on parts" is defined by COP = { f in C(m) I f is constant each Gleason part P(m) for all m in m\D}. AOP is important in determining compactness of the semi-commutator Tf Tg -

TTg in [4] and [14] and plays the same role on the Bergman space as the algebra H ~ + C(oqD) on the Hardy space to describle the compactness of certain semi-commutators of Toeplitz operators on the Hardy space ([3], [13]). So this motivates us to prove that AOP = AQN C('TP.). Theorem 6. For f in C(I"1~), then (a) Hf is compact iff f is in AOP,

(b) Both Hf and HT are compact iff f is in COP.

Proof. It suffices to prove (a) since AOPNAOP = COP. From Theorem 3 it follows that Hf is compact if and only if

lirn llfOqbz-P(fO(~z)ll2 =0 (11). Izl+ 1 So we have to show that (11) is equivalent to the statement that f is in AOP.

For every m in ~\D, the Corona Theorem says that there is a net { z~ in D converging to m. H1 implies that ~zo~(w) -> Lm(w) pointwise. Then foezcz(w) converges to foLm(w) pointwise and II foezoc I1= - Ilfll~. Therefore by the dominated Zheng 293 convergence theorem we get lim II fO~z - f~ 112 =0. z ..).m a O~ Since the Bergman projection is bounded, II P(f~ ) - P(f~ 112 = 0. z -)m Cr

Therefore lim II f~ -P(f~162 ) 112 = 0 z -)m a a if and only if

II f~ P(foLm) 112 = 0

This means that foL m = P(foLm). Thus foL m is in H =. Combining the Corona Theorem and the compactness of maximal ideal space ~ of H = in the weak-star topology implies that (11) is equivalent to the statement that f is in AOP.

Theorem 7. For f in C(~), if P(foLm) = 0 for all m in 'il~\D, then f~ m = 0 for all m in ~\D. To prove Theorem 7 we need the following lemma. Lemma 5. Let m(z) = Lm(z ) be in P(m) for some z in D. Then there is a constant c with 1cl = 1 such that

Lm(z)(cw ) = Lmo~z(W ).

Lemma 5 is a special case of Theorem X 2.5 in [8]. Now we come to the proof of Theorem 7. Proof of Theorem 7. First we want to prove that

for all g in H•(D) and for all m in ~\D.

Suppose that m is in 'mkD and there is a net z~ in D converging to m (by the

Corona Theorem). For a g in H ~176and z in D, h in H=(D) then 294 Zheng

0 = < P(foLm(z)), [goLm(z) - goLm(z)(0)]ho(~zi~z-1 ~ >

= < (f~ [g~ "g~176 -l~ >

= < [-g~ - g~ (T~ h~ ~ > (from Lemma 5 and assuming that c is 1 )

= < [g~ - g~ ( T~176 ) h~ ~ > ( from Lemma 4 )

= < [goLm - goL m (0)]1~z, (ToL m ) h >

= < (I- P)(goL m ~z), (T~ h >

= < H goLrn (~z), (T~ h >

= < H goLm ( ~z),HToLm(h) >

= < HT~ H goLm ('Rz), h >

Since H~176 is dense in La2(D ) we have HyoLm* H ~oLrn (~z) = 0. Thus

HToLm* H ~oLm = 0.

On the other hand for ~ > 0 it follows from H2 that there are fj and gj in H~ ( j = 0 to n ) such that

h-- II f- j~ofjgj I1~ < Thus

13 -- II f~ j~ofj~176 Iloo < So

IIH ToLrn HT~ ~=o~ Hf~ H g-j~176 Iloo< Ilfllooe Therefore

IIH 7~ H ToLmIL < Ilfll,~ e

Since ~ is arbitrary we get Zheng 295

H ~oLm* H ToLm = 0

This means that H foLm = 0. So ~oL m is in H ~176This implies that

P( foe m ) = foLm (0) = m(f).

Since m is arbitrary we have foL m (z) = 0 for all z in D and m in 111\D to complete the proof. Consequently we can prove the following result which first appeared in [12].

Corollary. For f in C(~11), Tf is compact if and only if f is continuously extendable to D with flaD = 0.

Proof. It follows from Theorem 4 that Tf is compact if and only if

,P (re,#, = o Izl+1 if and only if P( foL m ) = 0 for all m in 'III\D. Theorem 7 implies that P(foLm) = 0 for all m in ~\D iff foL m = 0. So we get that Tf is compact if and only if lira f(z) = O. Izl+ 1

Thus f may be continuously extended to D with f[aD = 0 if and only if Tf is compact.

To conclude the section we prove the following theorem.

Theorem 8. For f in AOP, then the essential spectrum ~e(Tf) of Tf is f(TI/kD). So it is connected. Proof: First we show that

f(~ll\D) c ~e(Tf).

Suppose ~ is in ~e(Tf). Then there is an m in 'rll\D such that ~,=f(m). From the Corona

Theorem there is a net {zet} in D converging to m. Since f~ m is in H'~(D) we have

T f-eLm -~ 1-- 0. On the other hand 296 Zheng

11r~o L .~ 1 II m

=lira IIP(T.,z - X ) II2 z § t~

=lira II UzP(f.~z -~) II2 z §

=lirn IIe(~~ - ~ ) II2 z ..)m ix O/ =lim IITT_X ~z 112 Z .-)m {z

--0

Therefore 3` is in ce(Tf).

Now we show that C\f(~kD) c C\ oe(Tf). Suppose 0 ~ 3, in Ckf(~\D). Then there is an r with l>r>0 such that 3` is not in f(D\rD). If a net {z(z } in D converges to m then

(f%D\rD "3`)-1 .r converges pointwise to ( foLm(w ) - 3`)-1. Thus

11 (f%D\rD - 3,)-l~162 P [ (f%D\rD - 3`)-1or ]11 -~ o as z(z + m. Therefore

II (f%D\rD- 3-)-1~162 P [ (f%D\rD" 3`)-1Or II + 0 as Izl ~ 1.

From Theorem 3 it follows that H (f%D\rD - 3`)-1 is compact. Since for any f, g in L~~ TTTg- TTg = Hf*Hg we have

T (f%ovD - 3`)-1 T (f%DxrO - 3`) = I + H (f~D\rD - ~)-1*H (f%D\rD - ~)

T (f~GD\rD - 3`) T (f%D\rD - 3`)-1 = I + H (f~faDkrD - ~)*H (f%D\rD - 3`)-1 SO

T (f. 3`) T (f%D\rD - 3` )-1= I + Tf%rD T (f~D\rD - 3` )-1 +

H ~%g\rO - ~)*H (f%g~rD - [)-1 Zheng 297

and

T (f~D~rD - ~)-1 T (f _ ~) = I + T (f~DVD - ~)-1 Tf%r o

H (7%D~D - ~)-I*H (f%DVD - ~.)

Since mf%rg , H (f%g\rO - X) and H (f%D\rD - ~,)-1 are compact, then T (f _ ~.) is a Fredholm

operator. Thus ~ is not in Cre(Tf). So far we have proved that

(~e(Tf) = f(Til\D). In fact

~e(Tf) = n(f(D\r'D): 1 >r>0}

is the intersection of a nested family of compact connected sets, so it is connected. The proof is complete. Section 5. The unit ball B n and polydisc D n in O n.

Let f~ be a bounded symmetric domain with its standard (Harish-Chandra) realization in C n. For dv, the usual Euclidean volume measure on Cn=R 2n,

normalized so that v(f~)=l. As in [5], [6] we can consider, for f in L~~ the Toeplitz

operator Tf and Hankel operator Hf on the Bergman space La2( f~,dv ).

It follows from [10] that for each a in fZ, there is a biholomorphic automorphism ~a

of f2 with the properties

(1) ~a(a) = 0,

(2) ~aO~a = identity map.

In [5] and [6] it was shown that ~a and the Bergman reproducing kernel on f~

have many similar properties to those of mobius transformations and the Bergman reproducing kernel on the unit disk D in C. By proposition 2.7 in Forelli and Rudin [ 7], an inequality on the unit ball IBn which is analogous to Lemma 1 in Section 2, and the property that the Bergman kernel of polydisc D n is the multiple product of Bergman 298 Zheng kernels of D, the results in Section 3 are valid on B n and D n .

We can now simply state our theorems for these domains and omit the proofs - which are almost the same as those presented in section 3.

Theorem 9. Let ~ be B n or Dn.For f in L~(~, dv ) the following are equivalent (9 .a) Hf is compact,

(9 .b) ~ IIHf 'h z II = O, z§ (9 .c) ]-~ II P(f,~z ) - fo~zllp = 0 for all p > 1.

Theorem 10. Let ~ be B n or D n. For f in L~176 dv ) the following are equivalent (10.a) Tf is compact,

(lO.b) ]i--m [ITf "hz ]J = O, z§ (10.c) ltrn II P(fo(~z)llp = 0 for all p _> 1. ~+~

After I finished this paper, I received two preprints by Karel Stroethoff on compact Hankel operators on the Bergman space. Theorem 3 in Section 3 and Theorem 9 in Section 5 were also obtained, but his method is different from ours. The present version of Sections 2 and 5 of this paper are however slightly altered from the original version and benefit from a criticism made by Dr. Stroethoff.

References

1. Sheldon Axler, Hankel operators on Bergman space, Linear and complex anaylsis problem book ( V. P. Havin, S. V. Hrussev, and N. K. Nikolski, eds), Lecture Notes in Math, V. 1043 Springer-Verlag, Berlin, 1984, pp. 262-263;

2. Sheldon Axler, The Bergman space, the Bloch space, and commutators of multiplication operators, Duke Math. J. 53(1986) 315-332; Zheng 299

3. Sheldon Axler, S.-Y. Chang and D. Sarason, Products of Toeplitz operators, Integral Equations and Operator Theory, 1 (1978), 285-309;

4. Sheldon Axler and P. Gorkin, Algebras on the disk and doubly commuting Toeplitz operators, preprint;

5. C. A. Berger, L. A. Coburn and K. H. Zhu, Function theory on Cartan domains and the Berezin-Toeplitz symbol calculus, to appear in American J. Math.;

6. C. A. Berger, L. A. Coburn and K. H. Zhu, BMO and the Bergman metric on bounded symmetric domains, preprint;

7. F. Forelli and W. Rudin, Projections on spaces of holomorphic functions in balls, Indiana Univ. Math. J. 24 (1974) 594- 602.

8. J. Garnett, Bounded analytic functions, Academic Press, 1982;

9. P. R. Halmos and V. S. Sunder, Bounded integral operators on L 2 spaces, Springer-Verlag, 1978;

10. S. Helgason, Differential geometry, Lie groups and symmetric spaces, Academic Press, 1982;

11. K. Hoffman, Bounded analytic functions and Gleason parts, Annals of Math. 86(1967), 74-111 ;

12. G. McDonald and C. Sundberg, Toeplitz operators on the disc, Indiana Univ. Math. J. 28(1979), 595-611;

13. A. L. Volberg, Two remarks concerning the theorem of S. Axler, S.-Y. Chang and D. Sarason, J. Operator theory 7(1982), 207-218;

14. D. Zheng, Hankel opertors and Toeplitz operators on the Bergman space, to appear in J. Functionat AnaIysis;

15. K. Zhu, VMO, ESV, and Toeplitz operators onthe Be~man space, T.A.M.S. V302. N2. August1987.

Department of Mathematics SUNY at Stony Brook Stony Brook, New York 11794 Submitted: June 6, 1988 Integral Equations 0378-620X/89/020300-0451,50+0.20/0 and Operator Theory (c) 1989 Birkh~user Verlag, Basel Vol. 12 (1989)

A TRACE ESTIMATE OF (T'T) p - (TT*)P

Ariyadasa Aluthge and Daoxing Xia*

A trace estimate in I3] of (T*T)P - (TT*)P h extended to the case 0 < p < } in this note.

Let )/ ben separable Hilbert space and/~()/) the algebra of bounded linear operators on g. For 1 < k < co, (Ck, I" Ik) denotes the Schatten-Von Neumann k-class. For T E /~()/), Rat(T) denotes the algebra of operators of the form r(T), where r is a rational function with poles off the spectrum o(T) of T. The multi- cyclicity re(T) of T is the least cardinal number of a set X C )/such that the closed linear span of Rat(T)x is )/. In f3], R. Curto, P. Muhly and D. Xia proved that if T E s _< p < 1 and the negative part of (T'T) p - (TT*) p is in the trace class, then

tr((T*T) p - (TT*) p) <_ lm(T + X)w2p(a(T + X)) (1)

for every X E tTzp, where plE)-- :P//pP-adpd0. 2 JJ B

This estimate is a generalization of Voiculescu's result 16] which is a generalization of the well-known Berger-Shaw Theorem [11. In the present note we extend the above result to the caze 0 < p < ~o First, we define the class Ck and a k-norm I" [k for 0 < k < 1. For 0 < k < 1, let Ck be the class of all compact operators X E r satisfying

Ixlk -- tr((X'XW <

Then (Ok, [-Ik) is a complete k-normed space [71. The aim of the present note is to prove the following.

* This work is supported in part by a NSF grant. Aluthge and Xia 301

THEOREM. If 0 < p < T e s and the negative part of (T'T) p - (TT*) p is in the trace class, then (1) holds good for every X e C2p.

For 0 < p < 1, the p-modulus of quassitrlangularity of T is defined by

%(T) = liminf ](I- P)TPI~ , PeP()l) where P(~) is the set of all finite rank projections in/:()/). This qp(.) is not a semi- norm. It satisfies qp(cT) = Iclqp(T) and %(T1 + T2) p < qp(T1) p + qp(T2) p. Another choice for p-modulus of quassitriangularity is qp(T)p which is a semi-p-norm. But here we choose this qr(T) for the reason that we only have to change a small part of the lemmas and the proofs from the case p > 1 to 0 < p _< 1.

LEMMA 1. For T E s and 0 < p < 1, we have

qp(T) < m(T)I/PIITII .

The proof of this lemma is similar to that of proposition 1 in [6]. (Cf. I5] as well}. We only have to notice that for 0 < k < 1, if T E Ck and rank T = 1, then IT]k = IITIIk and that IA $ BIk < IAIk + IBlk.

LEMMA 2. Let T e L(~l), 0 < p < ~, If the negative part of (T'T) p - (TT*) p is trace class, then

tr((T*T)P - (T*T*)p) < q2p(T) 2p.

The proof of this hmma is similar to that of Lemma 2.4 in [3], since

IABI2 , = tr((B*A*AB)P) for 0 < p < !2"

LEMMA 3. Let 0 < p < oo, T be a semi-hyponormal operators and GT be the principal function of T. If (T*T)P - (TT*)P is trace class, than

trCCT*T)' - (TT*F) = ff o'(T)

The proof of this lemma is similar to that in [2], [4], [8]. The following lemma is the key point to the proof of the theorem. 302 Aluthge and Xia

LEMMA 4. Let R > O, e be a sufficiently small positive number, 0 < p < 1 and F c {z E ~ : Izl <_ R - e} be a dosed subset. I] [l = {z E e : Izl _< R}\F, then there is an operator D on a Hilbert space such that

o(D) c n, re(D) = n, [IDII <_ R,

(D'D) p - (DD*) P is trace class and

tr((D*D) p - (DD*) p) > -~(w2p($1) - ~).

PROOF. Step 1 For r > 0 and e e r let D(e,r) = {z e ~: Iz - e[ < r} and A2(D(e,r)) be the Bergman space on D(e,r). Consider the multiplication operator W = W(c,r) on A2(D(e,r)), (W f)(z) = zf(z), f e A2(O(c,r)). It is obvious that W is subnormal and hence it is semi-hyponormal. Besides, re(W) = 1 and Ilwll < [el + r. Assume that r > [c[, then it is easy to show that (W*W)~ - (WW*)P is trace class. By lemma 3, we have

tr((W*W) p - (WW*) p) = W2p(D(e, r)). since the principal function Gw is the characteristic function of a(W) -- D(e, r). Step Choose a set of complex numbers ej, and r i > 0,j = 1,2,-.. ,m such that 0 r D(e,, ri),

D(ei,ri) ND(ek, rt~)=r for j~k, U~= 1 D(ey,ri) C 1~ and

Let Wj -- W(ej,rj), be the multiplication operator on A2(D(ej,rj)) and I7r = )-~'~j~--I (~Wj. Then it is easy to see that (tTV*I~V)p - (ITVITV*)P is trace class, a(P;r c n, I1~11 ~< R and

tr((W'WF~ - (##*F) = ;~,(uj=~D(,;,1 m ~j)) 1 Aluthge and Xia 303

Step 8. Construct operator D as the orthogonal sum of n copies of I~, then D satisfies all the conditions in Lemma 3 which completes the proof of the hmma. The rest of the proof of theorem is similar to the first part of the proof of Theorem 3.7 in [3] and we omit the details.

References

[ 1 ] Berger, C. and Shaw, B., Selfcommutators of multicyclic hyponormal opera- tors are always trace class, Bull. Amer. Math. Soc. 70(1973), 1193-1190.

[2] Carey, R. and Pincus, J., Mosaics, principal functions, and mean motion in yon Neumann algebras, Acta Math. 138(1977), 153-218.

[ 3 ] Curto, R. Muhly, P. and Xia, D., A trace estimate for p-hyponormal operators. Integral Equations and Operator Theory 6(1983), 507-514.

[4] Pincus J. and Xia, D., Mosaics and principal function of hyponormal and semi-hyponormal operators, Integral Equations and Operator Theory 4(1981), 134-150.

[5] Voiculescu, D., Some extensions of quassitriangularity, Rev. Roum. Math. Pures et Appl. 18(1973), 1303-1320.

[ 6 ] Voiculescu, D., A note on quassitriangutarity and trace-class self-commutators, Acta Sa. Math. (Szeged) 42(1980), 159-199.

[ 7 ] Zelazko, W., On the locally bounded and m-convex topological algebras., Stud. Math. 19(1900), 332-356.

[8] Xia, D., On locally bounded topological algebras, Scientia Sinica 13(1964), 375-390.

[ 9 ] Xia, D., Spectral Theory of Hyponormal Operators, (1983) Birkhguser Verlag, Basel, Boston, Stuttgart.

Department of Mathematics Vanderbilt University Nashville, TN 37235 U.S.A. Submitted: October 13, 1988