CEJM 3(4) 2005 654–665 Central European Journal of Mathematics

Integral representations of unbounded operators by infinitely smooth kernels

Igor M. Novitski˘ı∗ Institute for Applied Mathematics Far-Eastern Branch of the Russian Academy of Sciences, 92, Zaparina Street, 680 000 Khabarovsk, Russia

Abstract: In this paper, we prove that every unbounded linear operator satisfying the Korotkov-Weidmann characterization is unitarily equivalent to an integral operator in L2(R), with a bounded and infinitely smooth Carleman kernel. The established unitary equivalence is implemented by explicitly definable unitary operators.

Keywords: Closed linear operator, integral linear operator, Carleman kernel, characterization theorems for integral operators MSC (2000): 47G10, 45P05

1 Introduction and statement of the main result

The characterization theorems for integral operators in L2 spaces [4, 7] say that these operators are models for most linear operators in Hilbert spaces; and, therefore there is a series of important problems in operator theory, including the famous invariant subspace problem, which can be equivalently reformulated to the class of integral operators (see, e.g., Problem 6 in [8, § 3]). On the other hand, in order to be analytically usable, an integral model of an operator should have a kernel being tractable by means of analytical techniques, and our emphasis here is on the bounded, infinitely smooth Carleman kernels on R2, uniformly approximable by their restrictions to bounded rectangles, which are in turn perfect for applying classical methods of integral equation theory. The apparatus of Fredholm’s minors has been adapted for such type kernels in [13, 14]. In a sense, the present paper can be thought of as a continuation of [12], where, in

∗ E-mail: [email protected] I.M. Novitski˘ı 655

particular, it was proved that if a bounded linear operator is unitarily equivalent to an integral operator, then that operator is unitarily equivalent to an integral operator with a Carleman kernel of any given degree of smoothness. Here, this result extends from the bounded to the unbounded operators in the assumption, and is restricted from the finitely differentiable to the infinitely differentiable kernels in the conclusion. The new result can be expressed by saying that if a closed linear operator is unitarily equivalent to an integral operator with a Carleman kernel, then that operator is unitarily equivalent to an integral operator with an infinitely smooth Carleman kernel. To state the result more precisely we need to record some definitions, and some known results along these lines, for comparison. Throughout this paper, H is a complex, separable, infinite-dimensional Hilbert space

with norm ∥·∥H and inner product ⟨·, ·⟩H, and the symbols C, N, and Z, refer to the complex plane, the set of all positive integers, and the set of all integers, respectively. Let C(H) be the set of all closed, linear, densely defined operators in H, let R(H)

be the algebra of all bounded linear operators on H, and let Sp(H) be the Schatten-von Neumann p-ideal of compact linear operators on H [2, Chapter III, §7]. For an operator ∗ S in C(H), DS stands for a linear manifold that is the domain of S, and S for the adjoint

to S (w.r.t. ⟨·, ·⟩H). We let C0(H) denote the collection of all those operators S in C(H) ∗ for which there exists an orthonormal sequence {en} ⊂ DS∗ such that lim ∥S en∥H = 0. n→∞ Let R be the real line (−∞, +∞) equipped with the Lebesgue measure, and let R L2 = L2( ) be the Hilbert space of (equivalence classes of)∫ measurable complex-valued functions on R equipped with the inner product ⟨f, g⟩ = f(s)g(s) ds and the norm L2 R 1 ∥f∥ = ⟨f, f⟩ 2 . An operator T ∈ C(L ) is said to be integral if there exists a measurable L2 L2 2 2 function T : R → C, a kernel, such that, for each f ∈ DT , ∫ (T f)(s) = T (s, t)f(t) dt for almost every s in R. R

2 A kernel T on R is said to be Carleman if T (s, ·) ∈ L2 for almost every fixed s in R.

Every Carleman kernel, T , induces a Carleman function t from R to L2 by t(s) = T (s, ·)

for all s in R for which T (s, ·) ∈ L2.

The existence question for integral operators which are not in C0(L2) seems to be open at present (see Problem 1 in [8, § 5]). The question has the negative answer for

integral operators whose domains are L2 (bounded integral operators) and for integral operators whose kernels are Carleman (Carleman integral operators), so both the classes

indicated are included in C0(L2). On the other hand, a characterization theorem due to Korotkov [5] and Weidmann [18] asserts that unitary transformations can be constructed

to bring operators of C0(L2) to integral form, with the representing kernels being at least simply Carleman. Here is a two-space version of that theorem.

Proposition 1.1. If S ∈ C0(H), then there exists a unitary operator U : H → L2 such −1 that the operator T = USU (DT = UDS) is an integral operator with a Carleman kernel. 656 I.M. Novitski˘ı

The main point we are interested here is to try to adjust the unitary operators U in Proposition 1.1 so that the Carleman kernels of the output operators T are SK∞-kernels, which we are going to define later. Fix any non-negative integer m and impose on a Carleman kernel T the following smoothness conditions: (i) the function T and all its partial derivatives on R2 up to order m are in C (R2, C), (ii) the Carleman function t, t(s) = T (s, ·), and all its (strong) derivatives on R up to

order m are in C (R,L2). ∥·∥ Throughout this paper, C(X,B), where B is a Banach space (with norm B), denotes ∥ ∥ ∥ ∥ the Banach space (with the norm f C(X,B) = supx∈X f(x) B) of continuous B-valued functions defined on a locally compact space X and vanishing at infinity (i.e., given any ∈ ⊂ ∥ ∥ f C(X,B) and ε > 0, there exists a compact subset X(ε, f) X such that f(x) B < ε whenever x ̸∈ X(ε, f)). A function T that satisfies conditions (i), (ii) is called an SKm-kernel [12]. In addition, an SKm-kernel T is called a Km-kernel [9, 11] if the conjugate transpose function T ∗ (T ∗(s, t) = T (t, s)) is also an SKm-kernel, i.e., ∗ (ii) the Carleman function t∗, t∗(s) = T ∗(s, ·), and all its (strong) derivatives on R up

to order m are in C (R,L2). ∗ Note, incidentally, that Conditions (i), (ii), and (ii) , do not depend on each other in general, and that Condition (i) rules out the possibility for the nonzero SKm-kernels to be depending, for example, on only the difference or the sum of arguments; there are also less trivial examples of inadmissible dependences. The scale of the SKm(Km)-kernels can naturally be supplemented by infinitely smooth Carleman kernels, as follows.

Definition 1.2. We say that a function T is an SK∞(K∞)-kernel if T is an SKm(Km)- kernel for each non-negative integer m.

What follows is a thorough exposition in the chronological order of what has been published about integral representability of linear operators by Km- and SKm-kernels, whenever m is a priori fixed non-negative integer.

Proposition 1.3. Let m be a fixed non-negative integer. Then

(A) if for an operator S ∈ C0(H) there exist a dense in H linear manifold D and an

orthonormal sequence {en} such that ∗ {en} ⊂ D ⊂ DS ∩ DS∗ , lim ∥Sen∥H = 0, lim ∥S en∥H = 0, n→∞ n→∞ H → −1 then there exists a unitary operator Um : L2 such that T = UmSUm (DT = UmDS) is an integral operator with a Km-kernel (cf. [9], [10]),

(B) if for an operator family {Sα : α ∈ A} ⊂ C0(H) ∩ R(H) there exists an orthonor- mal sequence {en} such that lim sup ∥S∗e ∥ = 0, lim sup ∥S e ∥ = 0, →∞ α n H →∞ α n H n α∈A n α∈A I.M. Novitski˘ı 657

then there exists a unitary operator Um : H → L2 such that for each α ∈ A the operator −1 m Tα = UmSαUm (DTα = L2) is an integral operator with a K -kernel of Mercer type (cf. [11]),

(C) if for a countable family {Sr : r ∈ N} ⊂ C0(H)∩R(H) there exists an orthonormal

sequence {en} such that ∥ ∗ ∥ → → ∞ sup Sr en H 0 as n , r∈N

then there exists a unitary operator Um : H → L2 such that for each r ∈ N the operator −1 m Tr = UmSrUm (DTr = L2) is an integral operator with an SK -kernel (cf. [12]).

It should be mentioned that the converse of (A), as well as of the singleton case of (B), also holds, and is due to a characterization theorem by Korotkov [6] for the so-called bi-Carleman integral operators. One more detail on Proposition 1.3 is that the proof of (C) given in [12] is not direct, and goes by a circuitous route, via Statements (B) and (A). Now we are ready to state our main result, which aims at the desirable sharpening of the conclusion of Proposition 1.1.

Theorem 1.4. If S ∈ C0(H), then there exists a unitary operator U∞ : H → L2 such that −1 ∞ the operator T = U∞SU∞ (DT = U∞DS) is an integral operator with an SK -kernel.

The result also refines the singleton case of Statement (C) of Proposition 1.3 by yielding the deeper conclusion under the weaker hypothesis. The next section presents a direct proof of Theorem 1.4, a proof that, unlike that of (C), relies on none of the foregoing smoothing results. It should also be emphasized that the proof below defines in an explicit way that unitary operator whose existence Theorem 1.4 guarantees, by sending a concrete orthonormal basis into another one, no

spectral properties of S other than S ∈ C0(H) are used.

Remark 1.5. In our manuscripts [16, 15, 17], we demonstrated that with some adap- tations a technique, which we will develop in the next section, works just as well for restricting the conclusions of Statements (A), (B), and (C), to the case m = ∞, under the same assumptions on the input operators, the proofs are also independent of each other and direct. Thus, the integral representations of bounded and unbounded linear operators by means of SK∞-kernels and K∞-kernels are completely studied.

Section 2 concludes with a short note concerning a further development of Theo- rem 1.4.

2 Proof of Theorem 1.4

In order to make the proof less intricate, we break it up into three “algorithmic” steps. The first step is a geometric preparation for the next two steps. In this step we choose 658 I.M. Novitski˘ı

in a suitable way an orthonormal basis {fn} for H and split the operator S ∈ C0(H) to construct four auxiliary operators Q, J, Γ , and A. The second step uses the constructed

operators and the basis {fn} to give first a general description and then an explicit

example of an orthonormal basis {un} ⊂ L2 of infinitely smooth functions on R. The principal result of the step is a unitary operator from H to L2, which sends the basis {fn} onto the basis {un} and is suggested as U∞ in the theorem. The rest of the proof (Step 3) is a straightforward verification that the constructed unitary operator does indeed carry S onto an integral operator having an SK∞-kernel.

Step 1. Preparing

If S ∈ C0(H), then, by definition, there is an orthonormal sequence {ek} ⊂ DS∗ such that ∗ ∥S ek∥H → 0 as k → ∞. Choosing a subsequence we may and do assume that

∑ 1 ∗ 4 ∥S ek∥H < ∞ (1) k ∑ ∑∞ (the sum notation will always be used instead of the more detailed symbol ). k k=1 ⊥ Let H be the closed linear span of the ek’s, and let H be the orthogonal complement of H in H. Assume, with no loss of generality, that dim H⊥ = dim H = ∞. Prove that

H ⊂ DS∗ . (2) ∑ ∑ ∗ Indeed, if f = ⟨f, ek⟩H ek ∈ H, then, by (1), the series ⟨f, ek⟩H S ek converges in H; k k ∗ since S is closed, the vector f does belong to DS∗ . H H If E is the orthogonal projection of onto H and I is the identity operator{ } on , ⊥ ⊥ − ∗ ∗ then the subset (I E)DS of DS is a dense subset in H . Then choose ek to be an orthonormal basis for the subspace H⊥, with the property { } ⊥ ⊂ − ∗ ek (I E)DS , (3)

and let {fn} be any orthonormal basis for H such that { } { } { } ∪ ⊥ fn = ek ek . (4)

Split the operator S as follows

S = (I − E)S + ES. (5)

In view of (3) the first summand Q = (I − E)S admits the representation ∑ ⟨ ⟩ ∑ ⟨ ⟩ ⊥ ⊥ ∗ ⊥ ⊥ ∈ Qg = Qg, ek H ek = g, S ek H ek on each g DQ = DS. (6) k k Since E ∈ R(H), the adjoint of the second summand in (5) is given by (ES)∗ = S∗E, and is, by (2), in R(H). I.M. Novitski˘ı 659

∗ For each f ∈ DS∗ , let z(f) = ∥S f∥H + 1, and define an operator Λ ∈ R(H) by

∑ 1 ⟨ ⟩ Λ = ( ) ·, e⊥ e⊥. (7) kz e⊥ k H k k k

Since, for each f ∈ H, the series

∑ 1 ⟨ ⟩ ( ) f, e⊥ S∗e⊥ kz e⊥ k H k k k converges in H, it follows that the domain DS∗ includes the range of Λ, and therefore that the operator S∗Λ is in R(H). ∗ Now, use the basis {fn} to check that the operator J = S E is in S 1 (H), and that ∗ 4 the operator Γ = S Λ is in S2(H):

∑ 1 ∑ 1 ∑ 1 4 ∗ 4 ∗ 4 ∥Jfn∥H = ∥S Efn∥H = ∥S ek∥H < ∞, n n k ∑ ∑ ∑ ∗ ⊥ 2 2 2 ∗ 2 S ek H π ∥Γ fn∥H = ∥S Λfn∥H = ( ) < . 2 ∗ ⊥ 2 6 n n k k S ek H + 1

In particular, it follows that the set {sn} of the s-numbers of J has the property that

∑ 1 2 sn < ∞. (8) n ∑ If J = sn ⟨·, pn⟩H qn is the Schmidt decomposition for J, then the closedness of S makes n it possible to write, for every g ∈ DS, ∑ ∗∗ ∗ ∗ ∗ ESg = (ES) g = (S E) g = J g = sn ⟨g, qn⟩H pn, (9) n so that (5) becomes S = Q + J ∗. (10)

Define one more auxiliary operator A ∈ S1(H) by

∑ 1 4 A = sn ⟨·, pn⟩H qn, (11) n and apply (twice) the Schwarz inequality to infer that if ∥f∥H = 1 then √ ∑ 1 ∗ 1 1 2 2 8 4 ∥Af∥H = sn |⟨f, pn⟩H| = (J J) f ≤ ∥Jf∥H , H √ n ∑ (12) ∗ 1 ∗ 1 ∗ 1 2 2 8 4 ∥A f∥H = sn |⟨f, qn⟩H| = (JJ ) f ≤ ∥J f∥H . H n 660 I.M. Novitski˘ı

Step 2. Defining a unitary U∞

In this step, we construct a candidate for the desired unitary operator U∞ in the theorem.

Notation 2.1. If an equivalence class f ∈ L2 contains a function belonging to C(R, C), then we shall use [f] to denote that function.

For each h ∈ H, let

1 1 4 ∗ 4 ∗ d(h) = ∥Jh∥H + ∥J h∥H + ∥Γ h∥H , (13)

and note that the compactness of each of J and Γ , proved in the previous step, implies that

d(ek) → 0 as k → ∞. (14) { }∞ Take any orthonormal basis un n=1 for L2, with the properties: (i) (a) for each i and for each n ∈ N, the ith derivative, [un] , of [un] is in C(R, C) (here and throughout, the letter i is reserved for all non-negative integers), { } { }∞ { }∞ (b) the set un is the disjoint union of two sequences gk k=1 and hk k=1 such that if (i) (i) Hk,i = [hk] , Gk,i = [gk] , then, for each i, C(R,C) C(R,C) ∑ Hk,i < ∞, (15) ∑ k( ) ⊥ ∞ kz ek Hn(k),i < , (16) ∑k d(xk)(Gk,i + 1) < ∞, (17) k { }∞ { }∞ where n(k) k=1 is a strictly increasing sequence of positive integers, and xk k=1 is a subsequence of the sequence {ek}.

Example 2.2. A good example of such a basis {un} can be adopted from the wavelet theory, as follows. Let ψ be the Lemari´e-Meyer wavelet, ∫ 1 ıξ( 1 +s) [ψ](s) = e 2 sgn ξb(|ξ|) dξ (s ∈ R) (18) 2π R

with the bell function b being infinitely smooth and compactly supported on [0, +∞) (see, e.g., [1, § 4] or [3, Example D, p. 62] for details). Then [ψ] is of the Schwartz class S(R), so its every derivative [ψ](i) is in C(R, C). In addition, the “mother wavelet” ψ generates { } an orthonormal basis ψjk j, k∈Z for L2 by

j j ψjk = 2 2 ψ(2 · −k)(j, k ∈ Z).

In a completely arbitrary manner, rearrange the two-indexed set {ψjk}j, k∈Z into a { }∞ simple sequence so that it becomes un n=1; and, reveal that the latter has the property I.M. Novitski˘ı 661

∈ N (b). Indeed, if, in view of that rearrangement, un = ψjnkn whenever n , then, for each i,

(i) (i) ≤ ∈ N [un] = [ψjnkn ] DnAi for all n , C(R,C) C(R,C) where   j2 2 n if jn > 0, ( ) 1 2 |j | (i+ ) (i) Dn = 1 n Ai = 2 2 [ψ] .  C(R,C)  √ if jn ≤ 0, 2 { }∞ → −∞ If a strictly increasing sequence{ l(k) k=1} of positive{ integers} satisfies jl(k) as → ∞ { } ∞ ∞ { }∞ k , then split un into hk = ul(k) k=1 and gk = um(k) k=1, with m(k) k=1 = N \{ }∞ l(k) k=1, and observe that ∑ Dl(k) < ∞. (19) k Then, for each i, the sums in (15), (16), and (17), are bounded by ∑ ∑ ( ) ∑ { } ⊥ Ai Dl(k),Ai kz ek Dl(n(k)), and (Ai + 1) d(xk) max 1,Dm(k) , k k respectively, where the last-written two expressions can always be made finite by an { }∞ N { } { } appropriate choice of subsequences n(k) k=1 of and xk of ek (see (19), (14)). { } { } ⊥ ⊥ ∪ { }\{ } Let us return to the proof of the{ theorem.} Let xk = ek ( ek xk ), and { } { } ∪ ⊥ H → observe via (4) that fn = xk xk . Define a unitary operator U∞ : L2 on the basis vectors by setting

⊥ ∈ N U∞xk = hk,U∞xk = gk, for all k , (20)

in the harmless assumption that, for each k ∈ N,

⊥ U∞fk = uk,U∞ek = hn(k), (21)

where {n(k)} is just that sequence which occurs in (16).

Step 3. Verifying

−1 In this step, we are to verify that T = U∞SU∞ (DT = U∞DS) is an integral operator having an SK∞-kernel. For this purpose, in view of the splitting (10), it suffices to −1 ∗ −1 check that the operators P = U∞QU∞ (DP = DT ) and F = U∞J U∞ (DF = L2) are integral operators having SK∞-kernels. The verification of the latter properties goes by representing all pertinent functions as infinitely smooth sums of termwise differentiable series of infinitely smooth functions. Combine (6) with (21) to infer that ∑ ⟨ ⟩ ∗ P f = f, T hn(k) hn(k) for all f ∈ DP , (22) L2 k 662 I.M. Novitski˘ı where, by (7), ∑ ⟨ ⟩ ( ) ∑ ⟨ ⟩ ∗ ∗ ⊥ ⊥ ∗ ⊥ T hn(k) = S ek , fn H un = kz ek S Λek , fn H un n n ( ) ∑ ⟨ ⟩ ⊥ ⊥ ∗ ∈ N = kz ek ek ,Γ fn H un (k ), (23) n with the series convergent in L2. Prove that, for any fixed i, the series ∑ ⟨ ⟩ ⊥ ∗ (i) ∈ N ek ,Γ fn H [un] (s)(k ) n converge in the space C(R, C). Indeed, all these series are pointwise dominated on R by one series ∑ ∗ (i) ∥Γ fn∥H [un] (s) , n which converges uniformly in R because its component subseries

∑ ∑ ∥ ∗ ∥ (i) ∗ ⊥ (i) Γ xk H [gk] (s) , Γ xk H [hk] (s) k k are in turn dominated by the convergent series ∑ ∑ ∗ d(xk)Gk,i, ∥Γ ∥ Hk,i, k k respectively (see (20), (13), (17), (15)). For (23), the reasoning just given implies in turn that, for each k ∈ N, [ ] ( ) ∗ (i) ≤ ⊥ T hn(k) Cikz ek , (24) C(R,C) with a constant Ci independent of k. From (23), it also follows that ( ) ∗ ⊥ T hn(k) ≤ kz e ∥Γ ∥ (k ∈ N). (25) L2 k

2 If a function P : R → C and a Carleman function p : R → L2 are defined as ∑ [ ] [ ] ∗ P (s, t) = hn(k) (s) T hn(k) (t), k ∑ [ ] ∗ (26) p(s) = P (s, ·) = hn(k) (s)T hn(k) k whenever s, t ∈ R, then, for all non-negative integers i and j,

∂i+jP ∑ [ ] [ ] (s, t) = h (i) (s) T ∗h (j) (t), ∂si∂tj n(k) n(k) k dip ∑ [ ] (s) = h (i) (s)T ∗h , dsi n(k) n(k) k I.M. Novitski˘ı 663

because, in view of (24), (25), and (16), the series just displayed converge (and even 2 absolutely) in C (R , C) and C (R,L2), respectively. Thus, ∂i+jP ( ) dip ∈ C R2, C , ∈ C (R,L ) (27) ∂si∂tj dsi 2 whenever i and j are non-negative integers. From (25) and (16), it follows that the series (22) (viewed, of course, as a series with terms belonging to C(R, C)) converges and even absolutely in the C(R, C) norm, and therefore that its pointwise sum is nothing else than [P f]. On the other hand, the established properties of the series in (26) make it possible to write, for each temporarily fixed s ∈ R, the following chain of relations ⟨ ⟩ ∑ ⟨ ⟩ [ ] ∑ [ ] ∗ ∗ f, T hn(k) hn(k) (s) = f, hn(k) (s)T hn(k) L2 k k ∫ ( )L2 ∫ ∑ [ ] [ ] ∗ = hn(k) (s) T hn(k) (t) f(t) dt = P (s, t)f(t) dt R R k

whenever f is in DP . This together with (27) imply that P : DP → L2 is an integral operator with the SK∞-kernel P . The next thing is to observe that the inducing kernel of the integral operator F = ∗ −1 U∞J U∞ ∈ S 1 (L2) is the sum of the bilinear series 4 ( ) ∑ 1 ∑ 2 ∗ sn U∞A qn(s)U∞Apn(t) = snU∞pn(s)U∞qn(t) (28) n n in the sense of almost everywhere convergence on R2 (see (9), (11)). The functions used in (28) can be written as the series ∑ ∑ ∗ ∗ U∞Apk = ⟨pk,A fn⟩H un,U∞A qk = ⟨qk, Afn⟩H un (k ∈ N), n n

(i) ∗ (i) convergent in L2. Show that, for any fixed i, the functions [U∞Apk] ,[U∞A qk] (k ∈ N) make sense, are all in C(R, C), and their C(R, C) norms are bounded independent of k. Indeed, all the series ∑ ∑ ∗ (i) (i) ⟨pk,A fn⟩H [un] (s), ⟨qk, Afn⟩H [un] (s)(k ∈ N) n n are dominated by one series ∑ ∗ (i) (∥A fn∥H + ∥Afn∥H) [un] (s) n that converges uniformly on R, because it consists of the two uniformly convergent sub- series on R

∑ ∑ ( ) ∥ ∗ ∥ ∥ ∥ (i) ∗ ⊥ ⊥ (i) ( A xk H + Axk H) [gk] (s) , A xk H + Axk H [hk] (s) , k k 664 I.M. Novitski˘ı

dominated by the convergent series ∑ ∑ d(xk)Gk,i, 2 ∥A∥ Hk,i, k k respectively (see (13), (12), (17), (15)). 2 Now define a function F : R → C and a Carleman function f : R → L2 by

∑ 1 2 ∗ F (s, t) = sn [U∞A qn](s)[U∞Apn](t), n ∑ 1 (29) 2 ∗ f(s) = F (s, ·) = sn [U∞A qn](s)U∞Apn, n whenever s, t ∈ R (cf. (28)). Then, for all non-negative integers i, j and all s, t ∈ R,

i+j ∑ 1 ∂ F ∗ (i) (j) (s, t) = s 2 [U∞A q ] (s)[U∞Ap ] (t), ∂si∂tj n n n n i ∑ 1 d f ∗ (i) (s) = s 2 [U∞A q ] (s)U∞Ap , dsi n n n n

2 as the series just written converge in C (R , C) and C (R,L2), respectively, due to (8). Therefore, it follows that F is the SK∞-kernel of F .

In accordance with (10), we have T = P + F , and hence, for each f ∈ DT , ∫ ∫ ∫ (T f)(s) = P (s, t)f(t) dt + F (s, t)f(t) dt = (P (s, t) + F (s, t))f(t) dt R R R

for almost every s in R. Therefore T : DT → L2 is an integral operator, and that kernel T of T , which is defined by T (s, t) = P (s, t) + F (s, t) whenever s and t are in R, inherits the SK∞-kernel properties from its terms. Consequently, T has the SK∞-kernel T . The proof of the theorem is complete.

Remark 2.3. For further research, there is at least one challenging question: Let S ∈ e C0(H). Does it then follow that there exists a unitary operator U∞ : H → L2 such that e e−1 ∞ U∞SU∞ is an integral operator having an SK -kernel which is real-analytic together with its Carleman function? In our opinion, the question is not answered by the unitary operator (20) in general because the series (26), (29) may fail to represent real-analytic functions, in spite of the fact that the function (18) is the restriction of an entire function e on R. A much more analytic machinery should be used to construct such a unitary U∞, if the latter does indeed exist.

Acknowledgment

The author wishes to express his gratitude to the Mathematical Science Division of the Russian Academy of Sciences for its support of this research (grant N 04-1-OMH-079), and to the referees for their remarks. I.M. Novitski˘ı 665

References

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