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 integral 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 properties 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 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 [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, measure 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 equations 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 definition 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)). Now recall that a bounded linear operator U : H→ L2(Y,µ) is unitary if U 2 has range L (Y,µ) and Uf,UgL2(Y,µ) = f, gH for all f, g ∈H. In addition, a linear operator S : DS ⊂H→H is unitarily equivalent to a linear operator 2 2 2 T : DT ⊂ L (Y,µ) → L (Y,µ) if there is a unitary operator U : H → L (Y,µ) −1 such that T = USU , meaning that DT = UDS. INTEGRAL REPRESENTATIONS OF UNBOUNDED... 363

A characterization theorem for Carleman (resp. bi-Carleman) operators is as follows: A necessary and suﬃcient condition that an operator S ∈ C(H) be unitarily equivalent to a Carleman (resp., bi-Carleman) operator in L2(Y,µ) is that S belong to the class C0(H) (resp., C00(H)) (see Korotkov [8] and Weidmann [14] (resp., Korotkov [9])); in particular case when S = S∗ and H = L2(Y,µ) = L2(a, b), each of these two characterizations turns into a pio- neering characterization of self-adjoint Carleman operators in L2(a, b), given in 1935 by von Neumann [13]. In a concrete setting with the underlying measure space (Y,µ) being R with the Lebesgue measure, the proof of the characterization theorem for Carleman operators was adjusted so as to yield the following, inﬁnitely smoothing, result.

Proposition 3. If S ∈ C0(H), then S is unitarily equivalent to a Carleman operator T in L2, with an SK∞ kernel. This result was proved in [11], Theorem 1.4. In the present paper, our goal is to state and prove a sharpened version of Proposition 3 when we restrict ourselves from the largest class of Carleman representable operators, C0(H), to the largest class of bi-Carleman representable operators, C00(H). The version adds to the hypothesis on the input operator S but sharpens the conclusion regarding quality of the output kernel proﬁtably, and is as follows.

Theorem 4. If S ∈ C00(H), then there exists a unitary operator U : H→ 2 −1 L such that the operator T = USU (DT = UDS) is a bi-Carleman operator with a K∞ kernel. In comparison with the previous proposition, the result delivers, besides properties (i) and (ii) (see Definition 2), the beneficial property (iii) for the kernel of the unitarily transformed operator T , at the cost of strengthening the spectral assumption on the transformable operator S (cf. (1) and (2)). In no way could Theorem 4 be obtained as an immediate consequence, or a special case, of the statement of Proposition 3. Note also that the truth of the converse of Theorem 4 is immediate from the italicized characterization theorem for bi- Carleman operators above. The next section of the present paper is entirely devoted to an independent proof of Theorem 4. The proof is, moreover, direct in the sense that it provides a direct method for constructing that unitary operator U : H → L2 whose existence the theorem asserts. The method uses no spectral properties of S other than (2) to determine the action of U by specifying two orthonormal bases, of H and of L2, one of which is meant to be the image by U of the other, the basis for L2 may be chosen to be an infinitely smooth wavelet basis. 364 I.M. Novitskii

The result of Theorem 4 has recently been published without proof in [12, Theorem 1].

2. Proof of Theorem 4

The structure of the proof is as follows: first we employ the hypotheses (2) to decompose H into an orthogonal direct sum H = L⊕L⊥ of infinite-dimensional subspaces in such a way that if E is the orthogonal projection of H onto L then ∗ the operators SE, S E are in the Schatten-von Neumann ideal S1/4(H). Also, at this stage, an orthonormal basis {f1,f2,f3,... }⊂ DS ∩ DS∗ for H is formed from orthonormal bases of L and of L⊥. In the next step (Step 2) we use ∗ the norms of the vectors Sfk, S fk (k ∈ N) to give a general description of 2 an orthonormal basis {u1, u2, u3,... }, for L , of infinitely smooth functions on R; the description is accompanied by showing that the Lemarié-Meyer wavelet basis (see [1], [7]) can serve as a concrete example of that basis. Then we define a unitary operator U : H → L2 by sending in a suitable manner the basis {f1,f2,f3,... } onto the basis {u1, u2, u3,... }. The last step of the proof −1 2 (Step 3) consists entirely of proving that the operator T = USU : UDS → L is an integral operator with a K∞ kernel.

Step 1. If S ∈ C00(H), then, by deﬁnition, there is an inﬁnite sequence of orthonormal vectors ek (k ∈ N) belonging to a dense linear manifold D ⊂ DS ∩ DS∗ in H and satisfying the limit relations of (2). By dropping down to ∞ a subsequence of {ek}k=1, also denoted ek, assume that 1 1 4 ∗ 4 SekH + S ekH ≤ 1 (3) Xk (the sum notation k will always be used instead of the more detailed symbol ∞ ⊥ k=1). Let L be the closed linear span of the ek’s, and let L be the orthogonal P ⊥ complement of L in H. Assume, with no loss of generality, that dim L = P dim L = ∞. Prove that

L ⊂ DS ∩ DS∗ . (4)

Indeed, if f = f, ekH ek ∈ L, then, by (3), both the series f, ekH Sek, k k ∗ ∗ and f, ekHPS ek converge in H; since both S and S are closed,P the vector k f doesP belong to both DS and DS∗ . If E is the orthogonal projection of H onto L and I is the identity operator INTEGRAL REPRESENTATIONS OF UNBOUNDED... 365 on H, then it follows from (2) and (4) that

(I − E)D ⊂ (I − E) (DS ∩ DS∗ ) ⊂ DS ∩ DS∗ , and hence that the linear manifold (I − E)D, being dense in L⊥, contains an ⊥ ⊥ ⊥ ⊥ orthonormal basis e1 , e2 , e3 ,... for the subspace L such that ⊥ ⊥ ⊥ ∗ e1 , e2 , e 3 ,... ⊂ DS ∩ DS . (5)

Then let {f1,f2,f3,... }nbe any orthonormalo basis for H including all terms of ∞ ⊥ ∞ the sequences {ek}k=1 and ek k=1: ⊥ ⊥ ⊥ {f1,f2,f3, ...}= {e1, e2, e3, ...}∪ e1 , e2 , e3 , ... . (6) Note that the operators S, S∗ decompose as n o S = (1 − E)S + ES, S∗ = (1 − E)S∗ + ES∗; (7) and observe that the summands Q = (1 − E)S, Q = (1 − E)S∗ here admit the representations ⊥ ⊥ e ∗ ⊥ ⊥ Qf = Qf,e e = f,S e e , k H k k H k Xk D E Xk D E (8) Qg = Qg,e⊥ e⊥ = g,Se⊥ e⊥, k H k k H k Xk D E Xk D E ∗ on all f of DQ =e DS and one all g of DQe = DS , respectively, by virtue of (5). Also observe that the adjoints of two other summands in (7) are given by (ES)∗ = S∗E, (ES∗)∗ = SE because E ∈ R(H). Hence, and because of (4), ∗ both J = SE and J = S E are in R(H). Moreover, J, J ∈ S1/4(H) as 1 1 1 1 4 ∗ 4 4 ∗ 4 SEfneH + S EfnH = SekH + eS ekH < ∞ n X Xk (see (6), (3)). In particular, it follows that the singular values, sn, of J and the singular values, sn, of J constitute 1/2-summable sequences: 1 1 s 2 < ∞, s 2 < ∞. (9) e n n e n n X X If e J = sn ,pnH qn and J = sn , pnH qn (10) n n X X are the Schmidt representations of J and Je, then thee closednesse e of each of S and S∗ yields ∗ ∗ ∗∗ ∗ ∗ e ES f = (ES ) f = (SE) f = J f = sn f,qnH pn (f ∈ DS∗ ), (11) n X 366 I.M. Novitskii

∗ ∗∗ ∗ ∗ ESg = (ES) g = (S E) g = J g = sn g, qnH pn (g ∈ DS), n X so (7) becomes e e e e ∗ S = Q + J , S∗ = Q + J ∗. (12) Step 2. In this step, we constructe a candidatee for the desired unitary oper- 2 ator U : H → L in the theorem. For each f ∈ DS ∩ DS∗ and for each h ∈H, let ∗ z(f)= SfH + S fH , 1 1 1 1 ∗ 4 (13) 4 ∗ 4 4 d(h)= JhH + J hH + Jh + J h , H H and use (3) to write e 1 e 1 ∗ 4 ∗ 4 z(ek)= SekH + S ekH ≤ SekH + S ekH 1 1 1 1 (14) ∗ 4 4 ∗ 4 ∗ 4 ≤ SekH + J ekH + S ekH + J ek = d(ek), H for each k ∈ N. Moreover, the orthonormality of the e and the compactness e k of each of J and J guarantee that

d(ek) → 0 as k →∞. (15) e Notation. If an equivalence class f ∈ L2 contains a function belonging to C(R, C), then we shall henceforth use [f] to denote that function.

2 Take any orthonormal basis {u1, u2, u3,... } for L , 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 sequence {un}n=1 splits into two inﬁnite subsequences {gk}k=1 and ∞ {hk}k=1 satisfying {h1, h2, h3,... } = {u1, u2, u3,... } \ {g1, g2, g3,... } , (16) (i) (i) and such that if Hk,i := [hk] , Gk,i := [gk] then, for each i, C(R,C) C(R,C)

Hk,i < ∞, (17) Xk z (vk) Hm(k),i < ∞, (18) Xk ⊥ z ek Hn(k),i < ∞, (19) Xk INTEGRAL REPRESENTATIONS OF UNBOUNDED... 367

d(xk)Gk,i < ∞, (20) k ∞ ∞ X where {n(k)}k=1, {m(k)}k=1 are two inﬁnite subsequences of the sequence ∞ {n}n=1 associated with each other through {m(1),m(2),m(3),... } = N \ {n(1),n(2),n(3),... } , (21) ∞ ∞ ∞ and {xk}k=1, {vk}k=1 are two inﬁnite subsequences of {ek}k=1 related with each other by

{x1,x2,x3,... } = {e1, e2, e3,... } \ {v1, v2, v3,... } . (22)

It is to be noted that since (cf. (14)) z(vk) ≤ 1 for all k ∈ N, the requirement (18) becomes superﬂuous if condition (17) holds; we have recorded this requirement here for convenience of presentation only.

Example. A good example of a basis {u1, u2, u3,... } with the above properties can be adopted from the wavelet theory, as follows. Let ψ be the Lemari´e-Meyer wavelet, 1 1 [ψ] (s)= sgn ξ eıξ( 2 +s)b(|ξ|) dξ (s ∈ R) (23) 2π R Z with the bell function b being inﬁnitely smooth and compactly supported on [0, +∞) (see, e.g., [1, § 4] or [7, 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, 2 the “mother wavelet” ψ generates an orthonormal basis {ψαβ}α, β∈Z for L by α 2 α ψαβ = 2 ψ(2 −β) (α, β ∈ Z).

In a completely arbitrary manner, rearrange the two-indexed set {ψαβ}α, β∈Z ∞ into a simple sequence so that it becomes {un}n=1. To show that the latter has N property (b), suppose un = ψαnβn whenever n ∈ , in accordance with that rearrangement. It is easily veriﬁed then that, for each i, (i) (i) N [un] = [ψαnβn ] ≤ DnAi for all n ∈ , C(R,C) C(R,C) where 2 αn 2 2 if αn > 0, (i+1/2) (i) Dn := Ai := 2 [ψ] . αn/2 R C (2 if αn ≤ 0, C( , )

∞ ∞ If a subsequence {l(k)}k=1 of {l}l=1 satisﬁes αl(k) → −∞ as k →∞, then split ∞ ∞ ∞ {un}n=1 into hk := ul(k) k=1 and gk := ur(k) k=1, with N {r(1), r(2) , r(3),... }= \ {l(1) , l(2), l(3),... } , 368 I.M. Novitskii and observe that

Dl(k) < ∞. (24) Xk Then, for each i, the sums in (17), (18), (19), and (20), are bounded respectively by ⊥ Ai Dl(k), Ai Dl(m(k)), Ai z ek Dl(n(k)), and Xk Xk Xk Ai d(xk)Dr(k), Xk where the last-written two expressions can always be made ﬁnite by an appro- ∞ ∞ ∞ ∞ priate choice of subsequences {n(k)}k=1 of {n}n=1 and {xk}k=1 of {ek}k=1 (see (24), (15)). Let us return to the proof of the theorem. Observe, by (6), (22), (16), and (21), that

{f1,f2,f3,... } = {x1,x2,x3,... } ⊥ ⊥ ⊥ ∪ {v1, v2, v3,... }∪ e , e , e ,... , 1 2 3 (25) n o {u1, u2, u3,... } = {g1, g2, g3,... }

∪ hm(1), hm(2), hm(3),... ∪ hn(1), hn(2), hn(3),... , 2 and deﬁne a unitary operator U : H→ L on the basis vectors by setting ⊥ N Uxk = gk, Uvk = hm(k), Uek = hn(k) for all k ∈ . (26) It is convenient and harmless to assume, in addition, that, for each n ∈ N,

Ufn = un, Uyn = hn. (27) Step 3. This step of the proof is to prove that the unitary operator U defined −1 in (26) does indeed possess the property that T = USU (DT = UDS) is a bi-Carleman operator with a K∞ kernel. First, for this purpose, verify that the −1 four operators (see the decompositions (12)) P = UQU (DP = DT ), P = ∗ − ∗ − − 1 e ∗ ∗ 1 2 1 UQU (DP = DT = UDS ), F = UJ U (DF = L ), and F = U J U 2 ∞ e (DFe = L ), are all Carleman operators with SK kernels. The checking is straightforward,e and goes by representing all pertinent kernelse and Carlemane functions as infinitely smooth sums of termwise differentiable series of infinitely smooth functions as follows. INTEGRAL REPRESENTATIONS OF UNBOUNDED... 369

Combine (8) with (26) to infer that ∗ Pf = f, T hn(k) L2 hn(k) for all f ∈ DT = UDS, k X (28) ∗ ∗ P g = g,Thn(k) L2 hn(k) for all g ∈ DT = UDS , k X where e ∗ ⊥ T h = e ,Sfn un, n(k) k H n X D E (29) ⊥ ∗ T h = e ,S fn un (k ∈ N), n(k) k H n X D E with the series convergent in L2. Prove that, for any ﬁxed i, the series ⊥ (i) ⊥ ∗ (i) e ,Sfn [un] (s), e ,S fn [un] (s) (k ∈ N) k H k H n n X D E X D E converge in the norm of C(R, C). Indeed, all these series are pointwise dominated on R by one series ∗ (i) (SfnH + S fnH) [un] (s) , n X R which converges uniformly on because its component subseries (see (13), (26), (25)) (i) (i) z(xk) [gk] (s) , z(vk) hm(k) (s) , Xk Xk ⊥ (i) z e k hn(k) ( s) k X are in turn dominated by the series of (20), of (18), and of (19), respectively (see also (14), (22)). Whence it follows that, for each k ∈ N, ∗ (i) ∗ (i) T hn(k) ≤ Ci , T hn(k) ≤ Ci, (30) C(R,C) C(R,C) ∗ with constants C and C i independent of k. From (13) it also follows via the i unitarity of U that ∗ ⊥ ⊥ N T hn(k) L2 ≤ z ek , T hn(k) L2 ≤ z ek (k ∈ ). (31) If functions P , P : R2 → C and Carleman functions p, p : R → L2 are

e e 370 I.M. Novitskii deﬁned as ∗ P (s,t)= hn(k) (s) T hn(k) (t), k X P (s,t)= hn(k) (s) T hn(k) (t), k X (32) e ∗ p(s)= P (s, )= hn(k) (s)T hn(k), k X p(s)= P (s, )= hn(k) (s)T hn(k), k R X whenever s, t ∈ , then,e fore all non-negative integer i and j, ∂i+jP (s,t)= h (i) (s) T ∗h (j) (t), ∂si∂tj n(k) n(k) k X ∂i+jP (s,t)= h (i) (s) T h (j) (t), ∂si∂tj n(k) n(k) k e X dip (s)= h (i) (s)T ∗h , dsi n(k) n(k) k X djp (s)= h (j) (s)T h , dsj n(k) n(k) k X because, in view of (30), (17),e (31), and (19), the series just displayed converge (and even absolutely) in C(R2, C) and C(R,L2), respectively. Thus, ∂i+jP ∂i+jP dip djp , ∈ C(R2, C), , ∈ C(R,L2), (33) ∂si∂tj ∂si∂tj dsi dsj whenever i and j are non-negativee integers. Also, frome (31) and (19), it follows that the series of (28) (viewed, of course, as series with terms belonging to C(R, C)) converge and even absolutely in the C(R, C) norm, and therefore that their pointwise sums are none other than [Pf] and P g , respectively. On the other hand, the established properties of the seriesh of (32)i make it possible to write, for each temporarily ﬁxed s ∈ R, the followinge chains of relations

∗ ∗ f, T h 2 h (s)= f, h (s)T h n(k) L n(k) n(k) n(k) k * k + 2 X X L ∗ = hn(k) (s) T hn(k) (t) f(t) dt = P (s,t)f(t) dt, R R Z k ! Z X INTEGRAL REPRESENTATIONS OF UNBOUNDED... 371

g,Thn(k) L2 hn(k) (s)= g, hn(k) (s)T hn(k) k * k + 2 X X L

= hn(k) (s) T hn(k) (t) g(t) dt = P (s,t)g(t) dt, R R Z k ! Z X e whenever f is in DP and g is in DP . These and (33) imply thate P : DP → ∞ 2 e 2 L , P : DP → L are integral operators with the SK kernels P and P , respectively. e e Now deﬁne two auxiliary operators B, B ∈ S1(H) by (cf. (10)) 1 1 B = s 4 ,p q , B = s 4 , p q , (34) n n H n e n n H n n n X X and apply the Schwarz inequality to infere that if efH e= 1 thene ∗ ∗ b(f) := BfH + B fH + Bf + B f H H 1 1 2 2 2 2 = sn |f,pnH| + e sn |f,q neH| s n s n X X 1 1 2 2 2 2 + sn |f, pnH| + sn |f, qnH| s n s n (35) X X ∗ 1 ∗ 1 = e(J J)e8 f + (JJ e) 8 f e H H ∗ 1 ∗ 1 8 8 + J J f + J J f H H 1 1 1 1 ∗ 4 4 e ∗ e4 4 e e ≤ JfH + J fH + Jf + J f = d(f). H H Then observe that the inducing kernels of the integral operators F = UJ ∗U −1, ∗ e e −1 2 F = U J U of S1/4(L ) are the sums of the bilinear series

1 e e 2 ∗ sn UB qn(s)UBpn(t) = snUpn(s)Uqn(t) , n n ! X X (36) 1 ∗ 2 sn U B qn(s)UBpn(t) = snUpn(s)Uqn(t) , ! n n X X 2 in the sense ofe almoste everywheree ee convergencee on Re (seee (11), (34)). The functions used in these expansions can be written as the series ∗ ∗ UBpk = pk,B fnH un, UB qk = qk,BfnH un, n n X X 372 I.M. Novitskii

∗ ∗ UBpk = pk, B fn un, U B qk = qk, Bfn un H H n n X D E X D E 2 (i) all convergingee in Le . Showe that, for anye fixede i, thee functionse [UBpk] , (i) ∗ (i) ∗ (i) [UB qk] , UBpk , U B qk (k ∈ N) make sense, are all in C(R, C), and their Ch(R, C)i normsh are bounded i independent of k. Indeed, all the series ee ∗ e (ei) (i) pk,B fnH [un] (s), qk,BfnH [un] (s), n n X ∗ X (i) (i) pk, B fn [un] (s), qk, Bfn [un] (s) (k ∈ N) H H n n X D E X D E are dominated bye one series e e ∗ e ∗ (i) B fnH + BfnH + B fn + Bfn [un] (s) H H n X R that converges uniformly on , because e it is composed e of the two uniformly convergent subseries on R (see (26), (27)): (i) (i) b(yk) [hk] (s) , b(xk) [gk] (s) , k k X X where the first series is dominated by the series of (17) multi plied by 2(B + B), and the second by the series of (20), because of (35). Now define functions F , F : R2 → C and Carleman functions f, f : R → L2 bye 1 e 2 ∗ e F (s,t)= sn [UB qn] (s)[UBpn] (t), n X1 ∗ 2 F (s,t)= sn U B qn (s) UBpn (t), n X h 1 i h i e e 2 ∗ e f(s)= F (s,e)= sn [UBe qn] (s)UBpe n, n X1 ∗ 2 f(s)= F (s, )= sn U B qn (s)UBpn, n h i R X whenever s, t ∈ e(cf. (36)).e Then, fore all non-negativee e integersee i, j and all s, t ∈ R, ∂i+jF 1 (s,t)= s 2 [UB∗q ](i) (s)[UBp ](j) (t), ∂si∂tj n n n n X ∂i+jF 1 ∗ (i) (j) (s,t)= s 2 U B q (s) UBp (t), ∂si∂tj n n n n e X h i h i e e e ee INTEGRAL REPRESENTATIONS OF UNBOUNDED... 373

dif 1 (s)= s 2 [UB∗q ](i) (s)UBp , dsi n n n n X djf 1 ∗ (j) (s)= s 2 U B q (s)UBp , dsj n n n n h i e X 2 as the series just written convergee (and evene e absolutely)e ine C(R , C) and C(R, L2), respectively, due to (9). Therefore, it follows that F and F are the SK∞ kernels of F and of F , respectively. Now, since (12) implies T = P + F , T ∗ = P + F , it followse from the above e 2 ∗ 2 that both the operators T : DT → L and T : DT ∗ → L admit the integral representations, e e ∗ ∗ (Tf)(s)= T (s,t)f(t) dt (f ∈ DT ), (T g)(s)= T (s,t)g(t) dt (g ∈ DT ∗ ), R R Z ∗ Z where T = P + F , T = P + F are SK∞ kernels. Then (cf. Remark 1) ∗ ∗ T (s,t)= T (t,s) for all (s,t) ∈ R2; hence T (,t)= T (t, ) in the L2 sense for each ﬁxed t ∈ R, whiche showse conclusively that T is a K∞ kernel of T . The proof of the theorem is complete.

3. Concluding Remark

In virtue of Theorem 4 and Remark 1, one can conﬁne one’s attention (with no essential loss of generality) to second-kind integral equations with K∞ kernels. One of the main technical advantages of dealing with such kernels is that their restrictions to compact rectangles in R2 are quite amenable to the methods of the classical theory of ordinary integral equations, and approximate their origi- nal kernels with respect to norms C(R2,C) and C(R,L2). This, for instance, can be used directly to establish an explicit theory of spectral functions for any ′ Hermitian K∞ kernel (T (s,t) = T (s,t)) by a development conceptually the same as the one given by Carleman [2, pp. 25-51] for a symmetric Carleman kernel on [a, b] × [a, b] that induces an unbounded integral operator in L2(a, b), but is, by construction, the pointwise limit of its symmetric Hilbert-Schmidt sub- kernels satisfying the mean square continuity condition. We believe that, with respect to K∞ kernels, this Carleman’s line of development can be extended far beyond the restrictive assumption of a Hermitian kernel.

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