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International Journal of Pure and Applied Mathematics ————————————————————————– Volume 54 No

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International Journal of Pure and Applied Mathematics ————————————————————————– Volume 54 No International Journal of Pure and Applied Mathematics ————————————————————————– Volume 54 No. 3 2009, 359-374 INTEGRAL REPRESENTATIONS OF UNBOUNDED OPERATORS BY INFINITELY SMOOTH BI-CARLEMAN KERNELS Igor M. Novitskii Institute for Applied Mathematics Far-Eastern Branch of the Russian Academy of Sciences Dzerzhinskiy Street 54, Khabarovsk, 680 000, RUSSIA e-mail: [email protected] Abstract: In this paper, we establish that if a closed linear operator in a separable Hilbert space H is unitarily equivalent to a bi-Carleman integral operator in an appropriate L2(Y,µ), then that operator is unitarily equivalent to a bi-Carleman integral operator in L2(R), whose kernel T : R2 → C and two Carleman functions t(s) = T (s, ·), t′(s) = T (·,s) : R → L2(R) are infinitely smooth and vanish at infinity together with all partial and all strong derivatives, respectively. The implementing unitary operator (from H onto L2(R)) is found by direct construction. AMS Subject Classification: 47G10, 45P05, 47B33, 47B38 Key Words: closed linear operator, integral linear operator, Carleman in- tegral operator, bi-Carleman integral operator, characterization theorems for integral operators, linear integral equation 1. Introduction and the Main Result The present paper may hopefully prove interesting for researchers who are inter- ested in the development of the theory of non-compact, non-self-adjoint linear integral operators in L2 spaces (see [6], [10]). In applications of this theory, such Received: June 6, 2009 c 2009 Academic Publications 360 I.M. Novitskii as occur, for instance, in the theory of singular integral equations of the second kind, it is often desirable to have the kernel function with special properties that make its associated integral operator easier to work with. Of these proper- ties the classically inspired are, for example, those of being bounded, infinitely differentiable, and bi-Carleman (that is, square integrable in each variable sep- arately for almost all values of the other). Here we focus attention on the kernel properties just listed, and try to show that up to a unitary equivalence these properties are simultaneously satisfied. Precisely, the problem we study in the present paper is to establish the largest class of those closed linear operators S in an abstract separable Hilbert space H that can be transformed by a suitable unitary operator US (from H 2 R −1 2 R onto L ( )) into an integral operator T = USSUS generated in L ( ) by a bounded, infinitely smooth, bi-Carleman kernel on R2, or, more concretely, by the K∞ kernel to be defined in Definition 2 below. It will turn out, and it will be the principal result of this paper (Theorem 4 below), that the operators S so transformable constitute the class which is precisely the same as that which was characterized by Korotkov in [9] to resolve the similar problem when no additional analytic properties other than just being bi-Carleman are required of the measurable kernels of unitary equivalents. In order to explain in detail the content of our main result, we need some notations, terminology and preliminaries. Throughout this paper, H is a com- plex, 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 [5, 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 with respect to ·, ·H. We let C0(H) denote the collection of all those operators S in C(H) for which there exists an ∞ orthonormal sequence {ek}k=1 in H such that ∗ {e1, e2, e3,... }⊂ DS∗ , lim S ekH = 0, (1) k→∞ and we let C00(H) denote the subset of C0(H) consisting of all those operators S in C(H) for which there exist a dense linear manifold D in H and an orthonormal INTEGRAL REPRESENTATIONS OF UNBOUNDED... 361 ∞ sequence {ek}k=1 in H such that {e1, e2, e3,... }⊂ D ⊂ DS ∩ DS∗ , ∗ (2) lim SekH = 0, lim S ekH = 0. k→∞ k→∞ Let R be the real line (−∞, +∞) equipped with the Lebesgue measure, and let L2 = L2(R) be the Hilbert space of (equivalence classes of) measurable R complex-valued functions on equipped with the inner product f, gL2 = 1/2 2 C 2 R f(s)g(s) ds and the norm fL = f,fL2 . An operator T ∈ L is said to be integral if there exists a measurable function T : R2 → C, a kernel, such R that, for each f ∈ DT , (Tf)(s)= T (s,t)f(t) dt for almost every s in R. R Z A kernel T on R2 is said to be Carleman if T (s, ·) ∈ L2 for almost every fixed s in R. To each Carleman kernel T there corresponds a Carleman function t : R → L2 defined by t(s)= T (s, ·) for all s in R for which T (s, ·) ∈ L2. The Carleman kernel T is called bi-Carleman in case its conjugate transpose kernel ′ ′ T (T (s,t) = T (t,s)) is also a Carleman kernel. Associated with the conju- ′ gate transpose T of every bi-Carleman kernel T there is therefore a Carleman ′ function t′ : R → L2 defined by t′(s)= T (s, ·) (= T (·,s)) for all s ∈ R in R for ′ which T (s, ·) ∈ L2. With each bi-Carleman kernel T , we therefore associate the pair of Carleman functions t, t′ : R → L2, both defined, via T , as above. An integral operator whose kernel is Carleman (resp., bi-Carleman) is referred to as the Carleman (resp., bi-Carleman) operator. Remark 1. The notions of integral operator, Carleman operator, and bi-Carleman operator, acting in the Hilbert space L2(Y,µ) are defined similarly as above in the space L2 (see [6], [10]); here and throughout (Y,µ) denotes a measure space with a positive, σ-finite, separable, and not purely atomic, mea- sure µ. It follows from the general theory that if T is a bi-integral operator on L2(Y,µ), that is, both T and its adjoint T ∗ are integral operators defined on all 2 2 2 of L (Y,µ), then T belongs to C00 L (Y,µ) ∩ R L (Y,µ) (see, e.g., [6, The- orems 3.10, 15.11]). The bi-integral operators, on the other hand, are generally involved in second-kind integral equations in L2(Y,µ), as the adjoint equations to such equations are customarily required to be integral. Note that a bi-integral operator need not be bi-Carleman; and another whole class of not necessarily 2 bounded integral operators that is entirely contained in C00 L (Y,µ) are just bi-Carleman operators, as will be seen from the discussion below. With regard to these latter operators, it is also relevant to mention the fact (of use in proving 2 the main result, Theorem 4) that if both the operator T : DT → L (Y,µ) and 362 I.M. Novitskii ∗ 2 its adjoint T : DT ∗ → L (Y,µ) are Carleman operators, with the kernels T ∗ ∗ ′ and T , respectively, then T is a bi-Carleman operator, and T (s,t)= T (s,t) for (µ × µ)-almost all (s,t) ∈ Y × Y (see [10, Corollary IV.2.17]). The converse 2 is also true. If, that is, T : DT → L (Y,µ) is a bi-Carleman operator with the ∗ 2 kernel T , then its adjoint T : DT ∗ → L (Y,µ) is a Carleman operator with a ′ kernel equal almost everywhere to T . Therefore, for the same reason as the bi-integral operators, the bi-Carleman operators can intervene in integral equa- tions of the second kind with closed operators. For description of some other important properties which the C00(H) operators may possess, we refer to [4, Theorem 5], [3, Theorem 2.3], and [6, Theorems 15.17, 15.18]. Among all possible Carleman and bi-Carleman kernels on R2, the next defi- nition distinguishes special types of those which, together with their associated Carleman functions, are infinitely smooth and vanish at infinity, and to which we shall restrict our consideration from now on. Definition 2. A bi-Carleman kernel T : R2 → C is called a K∞ kernel [11] if it satisfies the three generally independent conditions: (i) the function T and all its partial derivatives on R2 of all orders are in C R2, C , (ii) the Carleman function t, t(s) = T (s, ·), and its (strong) derivatives, dit , on R of all orders are in C R,L2 , dsi ′ (iii) the Carleman function t′, t′(s) = T (s, ·) = T (·,s), and its (strong) dit′ derivatives, , on R of all orders are in C R,L2 . dsi If a Carleman kernel T : R2 → C satisfies the above two conditions (i) and (ii), then it is called an SK∞ kernel [11]. Throughout this paper, C(X,B), where B is a Banach space (with norm ·B ), denotes the Banach space (with the norm fC(X,B) = supx∈X f(x)B) of continuous B-valued functions defined on a locally compact space X and vanishing at infinity (that is, 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)).
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