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Integral Representations of Unbounded Operators by Infinitely Smooth Kernels

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Integral Representations of Unbounded Operators by Infinitely Smooth Kernels 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, x 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, x7]. 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 feng ⊂ DS∗ such that lim kS enkH = 0. n!1 Let R be the real line (−∞; +1) equipped with the Lebesgue measure, and let R L2 = L2( ) be the Hilbert space of (equivalence classes of)R measurable complex-valued functions on R equipped with the inner product hf; gi = f(s)g(s) ds and the norm L2 R 1 kfk = hf; fi 2 . An operator T 2 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 2 DT , Z (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; ·) 2 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; ·) 2 L2. The existence question for integral operators which are not in C0(L2) seems to be open at present (see Problem 1 in [8, x 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 2 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 SK1-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 k k k k the Banach space (with the norm f C(X;B) = supx2X f(x) B) of continuous B-valued functions defined on a locally compact space X and vanishing at infinity (i.e., given any 2 ⊂ k k f C(X; B) and " > 0, there exists a compact subset X("; f) X such that f(x) B < " whenever x 62 X("; 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 SK1(K1)-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 2 C0(H) there exist a dense in H linear manifold D and an orthonormal sequence feng such that ∗ feng ⊂ D ⊂ DS \ DS∗ ; lim kSenkH = 0; lim kS enkH = 0; n!1 n!1 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 fSα : α 2 Ag ⊂ C0(H) \ R(H) there exists an orthonor- mal sequence feng such that lim sup kS∗e k = 0; lim sup kS e k = 0; !1 α n H !1 α n H n α2A n α2A I.M. Novitski˘ı 657 then there exists a unitary operator Um : H! L2 such that for each α 2 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 fSr : r 2 Ng ⊂ C0(H)\R(H) there exists an orthonormal sequence feng such that k ∗ k ! ! 1 sup Sr en H 0 as n ; r2N then there exists a unitary operator Um : H! L2 such that for each r 2 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 2 C0(H), then there exists a unitary operator U1 : H! L2 such that −1 1 the operator T = U1SU1 (DT = U1DS) is an integral operator with an SK -kernel.
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