International Journal of Pure and Applied Mathematics ————————————————————————– Volume 56 No

International Journal of Pure and Applied Mathematics ————————————————————————– Volume 56 No. 2 2009, 265-280

SIMULTANEOUS INTEGRAL REPRESENTABILITY BY INFINITELY SMOOTH KERNELS WITH APPLICATION TO INTEGRAL EQUATIONS

Igor M. Novitskii Khabarovsk Division Institute for Applied Mathematics Far-Eastern Branch of the Russian Academy of Sciences 54, Dzerzhinskiy Street, Khabarovsk, 680 000, RUSSIA e-mail: [email protected]

Abstract: In this note, we characterize families incorporating those bounded linear operators on a separable Hilbert space that can be simultaneously H transformed by the same unitary equivalence transformation into Carleman integral operators on L2(R), whose kernels T : R2 C and Carleman functions → T (s, ): R L2(R) are infinitely smooth and vanish at infinity together with → all partial and all strong derivatives, respectively. An explicit procedure for constructing the unitary operators, from onto L2(R), effecting such trans- H formations is also presented. As an application, we present a smooth version of Korotkov’s reduction method for general third-kind integral equations in L2(Y,µ), whose aim is to obtain an equivalent integral equation of either the first or the second kind in L2(R), with an infinitely smooth Hilbert-Schmidt or Carleman kernel, respectively.

AMS Subject Classiﬁcation: 47G10, 45P05, 47B33, 47B38, 47N20 Key Words: bounded Hilbert space operator, Hilbert-Schmidt operator, bounded integral linear operator, Carleman integral operator, characterization theorems for integral operators, linear integral equation of the third kind

Received: September 7, 2009 c 2009 Academic Publications 266 Igor M. Novitskii

1. Introduction and the Main Result

Throughout the note, is a complex, separable, infinite-dimensional Hilbert H space with norm and inner product , , and the symbols C, N, and H H Z, refer to the complex plane, the set of all positive integers, and the set of all integers, respectively. R( ) denotes the Banach algebra of all bounded linear H operators on . For an operator A in R( ), A∗ denotes the Hilbert space H H adjoint of A. Throughout let (Y,µ) denote a measure space Y equipped with a positive, σ-finite, complete, separable, and not purely atomic, measure µ, and then let L2(Y,µ) denote the Hilbert space of (equivalence classes of) µ-measurable complex-valued functions on Y equipped with the inner product f, g L2(Y,µ) = 1/2 f(y)g(y) dµ(y) and the norm f 2 = f,f 2 . When µ is the Y L (Y,µ) L (Y,µ) Lebesgue measure on the real line R = ( , + ), we abbreviate L2(R,µ) R −∞ ∞ to L2, and dµ(y) to dy. An operator T R L2(Y,µ) is said to be integral if there is a µ µ- ∈ × measurable function T : Y Y C, a kernel, such that, for every f L2(Y,µ), × → ∈ (Tf)(x)= T (x,y)f(y) dµ(y) for µ-almost every x Y . ∈ ZY A kernel T on Y Y is said to be Carleman if T (x, ) L2(Y,µ) for µ-almost × ∈ every fixed x in Y . Every Carleman kernel, T , induces a Carleman function t from Y to L2(Y,µ) by t(x)= T (x, ) for all x in Y for which T (x, ) L2(Y,µ). ∈ An integral operator with a kernel T is called Carleman if T is a Carleman kernel. Recall that a bounded linear operator U : L2(Y,µ) is unitary if U 2 H→ has range L (Y,µ) and Uf,Ug 2 = f, g for all f, g . L (Y,µ) H ∈H This note is written in the spirit of the pioneering work by Korotkov [5] on the simultaneous reducibility of families of linear operators to integral form by means of unitary equivalence transformations and its applications to integral equations. In that paper, Korotkov showed that if operators B R( ) (γ ) γ ∈ H ∈ G with an index set of arbitrary cardinality satisfy the condition B∗v , lim sup γ n H = 0 (1) n→∞ γ∈G where vn is an orthonormal sequence in , then there exists a unitary oper- { } 2 H −1 ator U : L (Y,µ) such that, for each γ , the operator Tγ = UBγU is H→ 2 ∈ G an integral operator on L (Y,µ), (Tγ f)(x) = Y Kγ(x,y)f(y) dµ(y) where the kernel K is Carleman. Motivated by this result, we consider in this note the γ R existence of a similar result when we specialize L2(Y,µ) to be L2 and require SIMULTANEOUS INTEGRAL REPRESENTABILITY BY... 267

each Carleman kernel Kγ to fulfill some additional analytic conditions that are to be technically useful in many applications, including the solving of integral equations. More specifically, we prove that when in (1) is at most countable, G the operators Bγ can simultaneously be made to have Carleman kernels that are SK∞ kernels in the sense of the following definition. Definition 1. A Carleman kernel T : R2 C is called an SK∞ kernel [9] → if it satisfies the two generally independent conditions: (i) the function T and all its partial derivatives on R2 of all orders are in C R2, C , dit (ii) the Carleman function t, t(s)= T (s, ), and its (strong) derivatives, i , ds on R of all orders are in C R,L2 . An SK∞ kernel T is called a K∞ kernel [9] if the conjugate transpose ′ ′ function T (T (s,t)= T (t,s)) is also an SK∞ kernel, that is, if additionally ′ (iii) the Carleman function t′, t′(s) = T (s, ) = T ( ,s), and its (strong) ′ dit R R 2 derivatives, dsi , on of all orders are in C ,L . Throughout this note, C(X,B), where B is a Banach space (with norm ), denotes the Banach space (with the norm f = sup f(x) ) B C(X,B) x∈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) < ε whenever x X(ε, f)). ⊂ B ∈ We are now ready to state our main result.

Theorem 2. Suppose that for a family = Br : r N R( ) there ∞ B { ∈ } ⊂ H exists an orthonormal sequence vn n=1 in such that ∗ { } H sup Br vn H 0 as n . (2) r∈N → →∞ Then there exists a unitary operator U : L2 such that, for each r N, the −1 H→ ∞ ∈ operator UBrU is a Carleman operator with an SK kernel. This result has recently been published without proof in [10, Theorem 2]. For the case in which is a singleton, its proof can be found in [9, Theorem B 1.4]. In [6, Section 3], there is a counter-example to show that the conclusion of Theorem 2 cannot in general be extended to operator families of more than countable cardinality. Section 2 of the present note is entirely devoted to proving Theorem 2. There we provide a direct method for constructing that unitary operator U : L2 whose existence the theorem asserts. The method uses no spectral H → 268 Igor M. Novitskii

properties of the operators Br, other than their joint property imposed in (2), to determine the action of U by specifying two orthonormal bases, of and of H 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. Section 3 of the present note deals with an application of Theorem 2 that is aimed at implanting the SK∞ kernels into integral equations. There we follow Korotkov’s method [5], applied to take into account Theorem 2, and reduce the general linear integral equation of the third kind in L2(Y,µ) to an equivalent integral equation either of the second kind (Theorem 3) or of the ﬁrst kind (Theorem 4) in L2, with the kernel being the linear pencil of the SK∞ kernels or of the Hilbert-Schmidt K∞ kernels, respectively.

2. Proof of Theorem 2

The proof has two steps. The first step is to pick suitable orthonormal bases, f ,f ,f ,... for and u , u , u ,... for L2, and then to define a certain { 1 2 3 } H { 1 2 3 } unitary operator from onto L2 that carries f into u for each n, and is H n n suggested as U in the theorem. The second step is a straightforward verification that the constructed unitary operator is indeed as desired.

Step 1. Assume that sup N B 1. This is a harmless assumption, r∈ r ≤ involving no loss of generality; just replace B with B > 1 by 1 B . Pick r r Br r ∞ ∞ a subsequence, ek k=1, of the sequence vn n=1 of (2) such that { } { } 1 ∗ 4 M = sup Br ek H < (3) r∈N ∞ Xk (the sum notation k will always be used instead of the more detailed symbol ∞ ). For each r N, let k=1 P∈ 1 P S = B , Q = (1 E)S , J = S∗E, (4) r r r r − r r r where E is the orthogonal projection of onto the closed linear span of the H ek’s, and note that ∗ Sr = Qr + Jr . (5) Assume, with no loss of generality, that dim(1 E) = . Let e⊥, e⊥, e⊥,... − H ∞ 1 2 3 be any orthonormal basis for the subspace (1 E) , and let f1,f2,f3,... be − H { ∞ } any basis for including all terms of the sequences e ∞ and e⊥ : H { k}k=1 k k=1 f ,f ,f ,... = e , e , e ,... e⊥, e⊥, e⊥,... . (6) { 1 2 3 } { 1 2 3 }∪ 1 2 3 n o SIMULTANEOUS INTEGRAL REPRESENTABILITY BY... 269

It then follows from (4) and (3) that 1 1 1 2 4 ∗ 4 ∗ 4 Jrfn H Jrfn H = Sr ek H sup Sr ek H M, (7) ≤ ≤ N ≤ n n r∈ X X Xk Xk implying in particular that the operators J , J ∗ (r N) are all Hilbert-Schmidt. r r ∈ For each f , let ∈H 1 1 4 ∗ 4 d(f) = sup Jrf H + sup Jr f H + sup Γrf H , (8) r∈N r∈N r∈N where, for each r N, ∈ 1 ⊥ ⊥ Γr = ΛSr, with Λ = , ek ek . (9) k H Xk D E Clearly Λ, and hence each Γ , is a Hilbert-Schmidt operator on . Prove that r H d(e ) 0 as k . (10) k → →∞ For this, use again the standard facts on Hilbert-Schmidt operators (see, e.g., [2, Chapter III]) to write the two following chains of relations ∗ 2 ∗ 2 ∗ 2 2 sup Jr ek H Jr ek H |Jr |2 = |Jr|2 N ≤ ≤ r∈ r r r Xk X Xk X X = J f 2 = S∗e 2 r nH r kH (11) r n r k X X X X 8 2 1 ∗ 2 M π sup Br ek H , ≤ r2 N ≤ 6 r r∈ X Xk 2 2 2 ∗ 2 sup Γrek H Γrek H |Γr|2 = |Γr |2 N ≤ ≤ r∈ r r r Xk X Xk X X 2 4 (12) ∗ 2 1 ⊥ 1 1 π = Sr Λfn H Λek = = , ≤ r2 H r2 k2 36 r n r r X X X Xk X Xk where | | is the Hilbert-Schmidt norm. The truth of (10) is now immediate 2 from (7), (11), and (12). 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. Take any orthonormal basis u , u , u ,... for L2 that satisﬁes conditions: { 1 2 3 } (a) for each i and for each n N, ∈ [u ](i) C(R, C), (13) n ∈ (i) where [un] denotes the i-th derivative of [un] (here and throughout, the letter i is reserved for all non-negative integers), 270 Igor M. Novitskii

(b) the sequence u ∞ decomposes into two inﬁnite subsequences g ∞ { n}n=1 { k}k=1 and h ∞ satisfying { k}k=1 h , h , h ,... = u , u , u ,... g , g , g ,... , (14) { 1 2 3 } { 1 2 3 } \ { 1 2 3 } (i) (i) and such that if Hk,i = [hk] , Gk,i = [gk] then, for each i, C(R,C) C(R,C)

H < , (15) k,i ∞ Xk kH < , (16) n(k),i ∞ Xk d(x )G < , (17) k k,i ∞ Xk where n(k) ∞ is a subsequence of the sequence n ∞ , and x ∞ is a { }k=1 { }n=1 { k}k=1 subsequence of the sequence e ∞ . { k}k=1 What follows is an example showing the explicit existence of such a basis. Example. Let N be decomposed into two infinite subsequences l(k) ∞ { }k=1 and m(k) ∞ , and let u , u , u ,... be an orthonormal basis for L2 that { }k=1 { 1 2 3 } has property (13), and such that, for each i, (i) [un] NnDi (n N), (18) C(R,C) ≤ ∈ ∞ ∞ where Nn , Di are two sequences of positive reals, the first of which { }n=1 { }i=0 is subject to the restriction that N < . (19) l(k) ∞ Xk Then u , u , u ,... is a basis of the type described in (b) above. Indeed, { 1 2 3 } from conditions (18), (19) it follows that the requirement (15) may be satisfied for all i by taking h = u (k N), and hence that a subsequence n(k) ∞ k l(k) ∈ { }k=1 of n ∞ may be chosen so as to fulfill (16) for all i, with h = u { }n=1 n(k) l(n(k)) (k N). At the same time, on account of (10) and of (18), the sequence ∈ e ∞ does always have an infinite subsequence x ∞ that satisfies, for { k}k=1 { k}k=1 each i, the requirement (17) for g = u (k N). k m(k) ∈ In turn, an explicit example of a basis u , u , u ,... that enjoys the above { 1 2 3 } three properties (13), (18), and (19), can be adopted from the wavelet theory, as follows. Let ψ be the Lemarié-Meyer wavelet, 1 1 [ψ](s)= eıξ( 2 +s)sgn ξb( ξ ) dξ (s R) 2π R | | ∈ Z with the bell function b being infinitely smooth and compactly supported on [0, + ) (see [3, Example D, p. 62] for details). Then [ψ] is of the Schwartz ∞ SIMULTANEOUS INTEGRAL REPRESENTABILITY BY... 271 class (R), so its every derivative [ψ](i) is in C(R, C). In addition, the “mother S 2 wavelet” ψ generates an orthonormal basis ψαβ α, β∈Z for L by α { } ψ = 2 2 ψ(2α β) (α, β Z). αβ − ∈ In any manner, rearrange the two-indexed set ψ Z into a simple { αβ}α, β∈ sequence, so that it looks like u ∞ , clearly each term here, u , has property { n}n=1 n (13) for each i. Besides, it is easily verified that a norm estimate like (18) holds (i) for each [un] , and the factors in the right-hand side of (18) can be taken as 2 αn 2 if αn > 0, (i+1/2)2 (i) Nn = Di = 2 [ψ] , αn/2 R C (2 if αn 0, C( , ) ≤ with the convention that u = ψ (n N), in conformity with that rear- n αnβn ∈ rangement. This choice of N also gives N < (cf. (19)) whenever n k l(k) ∞ l(k) ∞ is a subsequence of l ∞ satisfying α as k . { }k=1 { }l=1 P l(k) → −∞ →∞ Returning to the proof of the theorem, let x⊥,x⊥,x⊥,... = e⊥, e⊥, e⊥,... ( e , e , e ,... x ,x ,x ,... ) , 1 2 3 1 2 3 ∪ { 1 2 3 } \ { 1 2 3 } wheren the subsequenceo nx ∞ of e o∞ is exactly that appeared in condition { k}k=1 { k}k=1 (b) above, and recall that by (6) and (14) ⊥ ⊥ ⊥ f1,f2,f3,... = x1,x2,x3,... x ,x ,x ,... , { } { }∪ 1 2 3 (20) u , u , u ,... = g , g , g ,... n h , h , h ,... o. { 1 2 3 } { 1 2 3 } ∪ { 1 2 3 } The time has come to construct a candidate for the desired unitary operator in the theorem. Define a unitary operator U : L2 on the basis vectors by H → setting Ux⊥ = h , Ux = g for all k N, (21) k k k k ∈ in the harmless assumption that, for each k N, ⊥ ∈ Ufk = uk, Uek = hn(k), (22) where n(k) ∞ is just that subsequence of n ∞ which occurs in condition { }k=1 { }n=1 (b) above. Step 2. This step of the proof is to show that, in fact, if the unitary operator U : L2 is defined as in (21), then the operators UB U −1 : L2 L2 (r N) H→ r → ∈ are all simultaneously Carleman operators with SK∞ kernels. Since scalar factors do not alter properties (i), (ii) of Definition 1, it is sufficient, in virtue of (4), to prove instead that each of the operators T = US U −1 (r N) is r r ∈ Carleman, and has an SK∞ kernel. For this in turn it suffices, by (5), to check −1 that the same properties are possessed by the operators Pr = UQrU , Fr = 272 Igor M. Novitskii

UJ ∗U −1 (r N). The checking is straightforward, and goes by representing all r ∈ pertinent kernels and Carleman functions as infinitely smooth sums of termwise differentiable series of infinitely smooth functions as follows. Fix, to begin with, an arbitrary index r N, and adopt the convention that ∈ operator notations will, from now on, be abbreviated by omitting the subscript r whenever it occurs. Note that the first summand Q in (5) admits the following representation ∗ ⊥ ⊥ Q = ,S ek ek , H Xk D E with respect the orthonormal basis e⊥, e⊥, e⊥,... for (1 E) (see also (4)). 1 2 3 − H Accordingly, the operator P , which is the transform by U of Q, may be written as (cf. (22)) Pf = f, T ∗h h (f L2), (23) n(k) L2 n(k) ∈ k X where, by (9), ∗ ∗ ⊥ ⊥ N T hn(k) = S ek ,fn un = k ek ,Γfn un (k ), (24) H H ∈ n n X D E X D E with the series convergent in L2. Prove that, for any fixed i, the series ⊥ (i) N ek ,Γfn [un] (s) (k ) H ∈ n X D E converge in the space C(R, C). Indeed, all these series are pointwise dominated on R by one series Γf [u ](i) (s) , nH n n XR which converges uniformly in because its component subseries (see (20), (21)) (i) ⊥ (i) Γxk H [gk] (s) , Γxk [hk] (s) H k k X X are respectively dominated by the series of (17) (see also (8)), and by the series of (15) multiplied by Γ . With respect to (24), the dominance argument just given yields in turn that, for each k N, ∈ ∗ (i) T hn(k) Cik, (25) C(R,C) ≤ with a constant Ci that is independent of k.

Introduce a function P : R2 C and a Carleman function p: R L2, → → SIMULTANEOUS INTEGRAL REPRESENTABILITY BY... 273 defined, for all s, t R, by ∈ ∗ P (s,t)= hn(k) (s) T hn(k) (t), k X (26) p(s)= P (s, )= h (s)T ∗h . n(k) n(k) k X Then again, the corresponding theorems on termwise differentiation of series imply that, 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 X dip (s)= h (i) (s)T ∗h , dsi n(k) n(k) k X in as much as the series just displayed converge (and even absolutely) in C R2, C and C R,L2 , respectively, due to (25) and (16). Thus, ∂i+jP dip C R2, C , C R,L2 (27) ∂si∂tj ∈ dsi ∈ whenever i and j are non-negative integers. From (25) and (16), it also follows that the series of (23), viewed 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 [Pf]. On the other hand, the established properties of the series of (26) make it possible to write, for each temporarily fixed s R, ∈ the following chain of relations

f, T ∗h h (s)= f, h (s)T ∗h n(k) L2 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 whenever f is in L2. This in conjunction with (27) implies that the operator P = UQU −1 : L2 L2 is a Carleman operator with the SK∞ kernel P . → Now, for the second summand in (5), write the Schmidt representation, J ∗ = s ,q p , (28) n nH n n X 1 where the s are the singular values (the eigenvalues of (JJ ∗) 2 ), p ,p ,p ,... n { 1 2 3 } and q ,q ,q ,... are orthonormal sets of the singular elements (the p are { 1 2 3 } n 274 Igor M. Novitskii

∗ ∗ eigenvectors for J J and the qn for JJ ). By (7) and Lemma XI.9.32 of [1] 1 2 sn < . (29) ∞ n X Introduce an auxiliary operator A R( ) by 1∈ H 4 A = sn ,p q , (30) nH n n X and note that, by the Schwarz inequality,

1 1 ∗ 2 2 2 2 Af + A f = sn f,p + sn f,q H H | nH| | nH| s n s n X X (31) 1 1 1 1 ∗ 8 ∗ 8 4 ∗ 4 = (J J) f + (JJ ) f Jf H + J f H d(f) H H ≤ ≤ if f = 1. According to (28) and (30), the kernel that induces the integral H Hilbert-Schmidt operator F = UJ ∗U − 1 on L2 is the sum of the bilinear series

1 2 ∗ sn UA qn(s)UApn(t) = snUpn(s)Uqn(t) , (32) n n ! X X in the sense of almost everywhere convergence on R2. The functions used in this bilinear expansion may be represented by the series UAp = p , A∗f u , UA∗q = q , Af u (k N) k k nH n k k nH n ∈ n n X X 2 (i) ∗ (i) convergent in L . Show that, for any ﬁxed i, the functions [UApk] , [UA 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 p , A∗f [u ](i) (s), q , Af [u ](i) (s) (k N) k nH n k nH n ∈ n n X X are dominated by one series ( A∗f + Af ) [u ](i) (s) . nH nH n n X R This series converges uniformly in , because it is composed of two uniformly convergent subseries on R (see (21), (22)) ∗ ⊥ ⊥ (i) A xk + Axk [hk] (s) , H H k X ( A∗x + Ax ) [ g ](i) (s) , kH kH k Xk where the ﬁrst series is dominated by the series of (15) multi plied by 2 A , and the second by the series of (17), on account of (31). SIMULTANEOUS INTEGRAL REPRESENTABILITY BY... 275

Now introduce a function F : R2 C and two Carleman functions f, ′ → f : R L2, defined by → 1 2 ∗ F (s,t)= sn [UA qn] (s)[UApn] (t), n X 1 2 ∗ f(s)= F (s, )= sn [UA q ] (s)UAp , n n (33) n X 1 ′ 2 ∗ f (t)= F ( ,t)= sn UA q [UAp ] (t), n n n X whenever s, t R (cf. (32)). Then, for all non-negative integers i, j and all s, ∈ t R, ∈ ∂i+jF 1 (s,t)= s 2 [UA∗q ](i) (s)[UAp ](j) (t), ∂si∂tj n n n n X ′ dif 1 djf 1 (s)= s 2 [UA∗q ](i) (s)UAp , (t)= s 2 UA∗q [UAp ](j) (t), dsi n n n dtj n n n n n X X as the series just written converge (and even absolutely) in C R2, C and in C R,L2 , respectively, due to (29). It follows that F satisfies parts (i), (ii), and (iii), of Definition 1, and therefore is a Hilbert-Schmidt K∞ kernel of F . In accordance with (5), T = P + F , and hence, for each f L2, ∈ (Tf)(s)= (P (s,t)+ F (s,t))f(t) dt (34) R Z for almost every s in R. Therefore T is a Carleman 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, the −1 ∞ Carleman operator T (= USrU ) has the SK kernel T . Upon recalling that r was arbitrary, this finally implies that all the operators T = US U −1 (r N) r r ∈ are Carleman operators having SK∞ kernels. That completes the proof of the theorem.

3. Application to Third-Kind Integral Equations

Throughout this section, the measure space (Y,µ) is nonatomic (that is, every µ-measurable set E Y can be divided into two disjoint subsets of equal ⊂ measure). The general integral equation of the third kind in L2(Y,µ) is an 276 Igor M. Novitskii equation of the form H(x)φ(x) λ K(x,y)φ(y) dµ(y)= ψ(x) for µ-almost all x Y , (35) − ∈ ZY where H : Y C (the coeﬃcient of the equation) is a given bounded µ- → measurable function, K : Y Y C (the kernel of the equation) is a given × → µ µ-measurable function inducing an integral operator K R L2(Y,µ) by × ∈ (Kφ)(x) = K(x,y)φ(y) dµ(y), the scalar λ of C is given, the function ψ of Y L2(Y,µ) is given, and the function φ of L2(Y,µ) is to be determined. R Theorem 3. Suppose that the essential range of the coeﬃcient H in (35) contains the point α C, that is, ∈ µ y Y : H(y) α < ε > 0 for all ε> 0. (36) { ∈ | − | } Then equation (35) is equivalent (via a unitary transformation) to a second-kind integral equation in L2, of the form

αf(s)+ (T 0(s,t) λT (s,t)) f(t) dt = g(s) for almost all s R, (37) R − ∈ Z 2 where the function f of L is to be determined, and both the functions T 0 and T are SK∞ kernels not depending on λ. Proof. The proof relies primarily on the following observation by Korotkov [5, Corollary 1]: If H is the multiplication operator induced on L2(Y,µ) by the coefficient H, and I is identity operator on L2(Y,µ), then the two-element family = B = H αI,B = K of bounded operators on = L2(Y,µ) sat- B { 1 − 2 } H isfies the assumptions of Theorem 2. The construction of Korotkov’s sequence v fulfilling (2) for this is likely to be of practical use and deserves to be { n} B expounded in some detail. If E Y is a µ-measurable set of positive finite measure, the orthonormal ⊂ sequence of generalized Rademacher functions with supports in E will be de- noted by R ∞ and is constructed iteratively through successive bisections { n,E}n=1 of E as follows: χ χ 1 R = E1 − E2 provided E E = E with µE = µE = µE, 1,E √µE 1 ⊔ 2 1 2 2

χE1 1 χE1 2 + χE2 1 χE2 2 R = , − , , − , 2,E √µE 1 provided E E = E with µE = µE for i, k = 1, 2, i,1 ⊔ i,2 i i,k 4 SIMULTANEOUS INTEGRAL REPRESENTABILITY BY... 277

χE1 1 1 χE1 1 2 + χE1 2 1 χE1 2 2 + χE2 1 1 χE2 1 2 + χE2 2 1 χE2 2 2 R = , , − , , , , − , , , , − , , , , − , , 3,E √µE 1 provided E E = E with µE = µE for i, k, j = 1, 2, i,k,1 ⊔ i,k,2 i,k i,k,j 8 and so on indefinitely; here χZ denotes the characteristic function of a set Z and the unions are disjoint. A relevant result due to Korotkov states that ∗ lim K Rn,E L2(Y,µ) = 0 (38) n→∞ for any integral operator K R L2(Y,µ) and any µ-measurable E Y with ∈ ⊂ 0 <µE < (see, e.g., the proof of Theorem 3 in [4]). ∞ Let Y ∞ be an ascending sequence of sets of positive finite measure, such { n}n=1 that Y Y , let ε ∞ be a sequence of positive reals strictly decreasing to n ↑ { n}n=1 zero, and define E = Y y Y : ε < H(y) α ε whenever n N. n n ∩ { ∈ n+1 | − |≤ n} ∈ Due to the assumption (36), one can always make the sets En to have finite nonzero measures by an appropriate choice of Y and ε (n N). Having done n n ∈ so, let v = R , where, for each n N, k is an index satisfying n kn,En ∈ n ∗ ∗ 1 B v 2 = K R 2 (39) 2 nL (Y,µ) kn,En L (Y,µ) ≤ n (cf. (38)). Since the En’s are pairwise disjoint, the vn’s form an orthonormal 2 sequence in L (Y,µ). Moreover, by construction of sets En, ∗ 2 1 2 2 B v 2 = H(y) α dµ(y) ε . (40) 1 nL (Y,µ) µE | − | ≤ n n ZEn ∗ One can now assert from (39), (40) that B v 2 0 as n for r nL (Y,µ) → → ∞ r = 1, 2. By Theorem 2, there is then a unitary operator U : L2(Y,µ) L2 such −1 −1 → that the operators T0 = UB1U , T = UB2U are both integral operators with SK∞ kernels. This unitary operator can also be used to transform the integral equation (35) into an equivalent integral equation of the form (37) ∞ in such a way that T 0 and T are just those SK kernels that induce T0 and T , respectively. In operator notation, such a passage from (35) to (37) looks as follows: Uψ = U(H λK)U −1Uφ = U(αI + B λB )U −1Uφ = − 1 − 2 αf + (T λT )f = g where f = Uφ, g = Uψ. The theorem is proved. 0 − Theorem 4. If, with the notation and hypotheses of Theorem 3, α = 0, then equation (35) is equivalent to a first-kind integral equation in L2, of the form

(Γ 0(s,t) λΓ (s,t)) f(t) dt = w(s) for almost all s R, (41) R − ∈ Z 2 where the function f of L is to be determined, and both the functions Γ 0 and Γ are Hilbert-Schmidt K∞ kernels not depending on λ. 278 Igor M. Novitskii

Proof. In this case the equation (37), equivalent to (35), becomes

(T 0(s,t) λT (s,t)) f(t) dt = g(s) for almost all s R. (42) R − ∈ Z Let m L2 be such that [m] is an inﬁnitely diﬀerentiable, positive function ∈ all whose derivatives [m](i) belong to C(R, R), and let M be the multiplication operator induced on L2 by m. Multiply both sides of equation (42) by m, to recast it into an equivalent equation of the form (41), with the same sought-for function f L2, the new right side w = Mg L2, and the new kernel Γ λΓ , ∈ ∈ 0 − where Γ 0(s,t) = [m](s)T 0(s,t), Γ (s,t) = [m](s)T (s,t). It is to be proved that ∞ Γ 0, Γ are Hilbert-Schmidt K kernels. The proof is further given only for Γ , as the proof for the other kernel Γ 0 is entirely similar. If t is the associated Carleman function of the SK∞ kernel T , then

2 2 2 2 2 Γ (s,t) dt ds = m (s) t(s) L2 ds t C(R,L2) m L2 < , R R | | R ≤ ∞ Z Z Z implying that Γ is a Hilbert-Schmidt kernel and hence does induce two Car- leman functions γ, γ′ : R L2 by γ(s) = Γ (s, ), γ′(t) = Γ ( ,t). The series → representation of T (see (34), (26), and (33)) gives rise to a series representation of Γ , namely, with the notation of the proof of Theorem 2, 1 ∗ 2 ∗ Γ (s,t)= Mhn(k) (s) T hn(k) (t)+ sn [MUA qn] (s)[UApn] (t) k n X X for all (s,t) R2. Moreover, for all non-negative integers i, j and all s, t R, ∈ ∈ ∂i+jΓ (s,t)= Mh (i)(s) T ∗h (j) (t) ∂si∂tj n(k) n(k) k (43) X 1 2 ∗ (i) (j) + sn [MUA qn] (s)[UApn] (t), n X diγ 1 (s)= Mh (i) (s)T ∗h + s 2 [MUA∗q ](i) (s)UAp , (44) dsi n(k) n(k) n n n k n ′ X X djγ 1 (t)= Mh T ∗h (j) (t)+ s 2 MUA∗q [UAp ](j) (t), (45) dtj n(k) n(k) n n n k n X X in as much as the series in the right sides of these equalities converge absolutely in C(R2, C), as regards (43), and in C(R,L2), as regards (44) and (45); their SIMULTANEOUS INTEGRAL REPRESENTABILITY BY... 279 dominant series are respectively:

i 1 i i 2 (j) i ∗ (r) Cj kHn(k),r Cr, sn [UApn] Cr [UA qn] , C(R,C) C(R,C) r=0 n r=0 Xk X X X i 1 i i 2 i ∗ (r) T Hn(k),r Cr, sn A Cr [UA qn] , C(R,C) k r=0 n r=0 X X X 1 X 2 ∗ −1 (j) Cj m L2 kHn(k),0, sn MUA U [UApn] , C(R,C) n Xk X i i (i−r) where Cr = r [m] C(R,R) (see also (25), (16), and (29)). Therefore ′ ∂i+j Γ R2 C diγ dj γ R 2 i j C , , i , j C ,L for all non-negative integers i, j, ∂s ∂t ∈ ds dt ∈ implying that Γ is a K∞ kernel. The theorem is proved. Remark. To solve equation (41), Picard’s theorem on the solvability of ﬁrst-kind integral equations can be applied; in addition, Schmidt’s singular values and elements, needed by the theorem to write down explicitly the solution to that equation, can be determined in terms of Fredholm formulae (see [7], [8], and [10], for details).

References

[1] N. Dunford, J.T. Schwartz, Linear Operators, Part II, Interscience Pub- lishers, New York, London (1963).

[2] I.Ts. Gohberg, M.G. Kre˘ın, Introduction to the Theory of Linear Non- Selfadjoint Operators in Hilbert Space, Nauka, Moscow (1965).

[3] E. Hern´andez, G. Weiss, A First Course on Wavelets, CRC Press, New York (1996).

[4] V.B. Korotkov, Systems of integral equations, Sibirsk. Mat. Zh., 27 (1986), 121-133; translated from Russian: Siberian Math. J., 27 (1986), 406-416.

[5] V.B. Korotkov, On the reduction of families of operators to integral form, Sibirsk. Mat. Zh. 28 (1987), 149-151; translated from Russian: Siberian Math. J., 28 (1987), 470-472.

[6] I.M. Novitskii, A note on integral representations of linear operators, In- tegral Equations Operator Theory, 35 (1999), 93-104. 280 Igor M. Novitskii

[7] I.M. Novitskii, Fredholm minors for completely continuous operators, Dal’nevost. Mat. Sb., 7 (1999), 103-122.

[8] I.M. Novitskii, Fredholm formulae for kernels which are linear with respect to parameter, Dal’nevost. Mat. Zh., 3 (2002), 173-194.

[9] I.M. Novitskii, Integral representations of unbounded operators by in- ﬁnitely smooth kernels, Cent. Eur. J. Math., 3 (2005), 654-665.

[10] I.M. Novitskii, Unitary equivalence to integral operators and an application, Int. J. Pure Appl. Math., 50, No. 2 (2009), 295-300.