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.