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A NOTE ON ANALYTICAL REPRESENTABILITY OF MAPPINGS

arXiv:math/9303205v1 [math.FA] 29 Mar 1993A NOTE ON ANALYTICAL REPRESENTABILITY OF MAPPINGSINVERSE TO INTEGRAL OPERATORSM.I.OSTROVSKIIAbstract. The condition onto pair (F, G) of function Banach spaces under whichthere exists a integral operator T : F →G with analytic kernel such that the inversemapping T −1 :imT →F does not belong to arbitrary a priori given Borel (or Baire)class is found.We begin with recalling some definitions [4, §31.IX].

Let X and Y be metric spaces.By definition, the family of analytically representable mappings is the least family ofmappings from X to Y containing all continuous mappings and being closed with respectto passage to pointwise limit.This family is representable as an union ∪α∈ΩΦα, where Ωis the set of all countableordinals and Φα are defined in the following way.1. The class Φ0 is the set of all continuous mappings.2.The class Φα(α > 0) consists of all mappings which are limits of convergentsequences of mappings belonging to ∪ξ<αΦξ.A mapping f : X →Y is called of α Borel class if the set f −1(F) is a Borel set ofmultiplicative class α for every closed subset F ⊂Y.It is known [1], [11], if Y is a separable Banach space, then the class Φα coincideswith the α Borel class for finite α and with α + 1 Borel class for infinite α.

In addition[12], if Y is a Banach space, then the class of all regularizable mappings from X to Y ,imortant in the theory of improperly posed problems coicides with Φ1.The present paper is devoted to the following problem. Let F and G be Banachfunction spaces on [0,1] and T : F →G be an injective integral operator with analytickernel.

What class of analytically representable functions can the mapping T −1 : TF →F belong to?Let us give some clarifications.1. By analytic kernel we mean the mapping K : [0, 1]×[0, 1] →C with the followingproperty: For some open subsets Γ and ∆of C such that [0, 1] ⊂Γ and [0, 1] ⊂∆,there exists an analytic continuation of K to Γ × ∆.2.

If the spaces are real then we consider mapping K taking real values on [0, 1] ×[0, 1].The main result of the present paper states that for wide class of pairs (F, G) andevery countable ordinal α there exists an injective integral operator T : F →G withan analytic kernel such that T −1 : TF →F does not belong to Φα.This result is a generalization of the Menikhes’ result [6]. The last states that thereexists an integral operator from C(0, 1) to L2(0, 1) with infinite differentiable kerneland nonregularizable inverse.

A.N.Plichko [9] generalized this result onto wide classesof function spaces. The relations of our results with the results of [9] will be discussedin remark 3 below.We use standard Banach space terminology and notation, as may be found in [5].For a subset A of a Banach space X by clA, linA and A⊥we shall denote the closureof A in the strong topology, the linear span of A and {x∗∈X∗: (∀x ∈A)(x∗(x) = 0)}respectively.

For a subset A of X by A⊤we shall denote {x ∈X : (∀x∗∈A)(x∗(x) =1

0)}. By w∗−lim we shall denote the limit in the weak∗topology.In the present paper we restrict ourselves to the case when F is a separable Ba-nach function space on the closed interval [0,1] continuously and injectively embeddedinto L1(0, 1).

It is clear that classical separable function spaces satisfy this condition.The case of nonseparable spaces continuously and injectively embedded into L1(0, 1) isdiscussed in remark 4 below.Let Γ be a bounded open subset of the complex plane such that Γ ⊃[0, 1]. Letf : Γ →C be an analytic function continuous on the closure of Γ .

(If we consider realspaces then we assume in addition that f takes real values on the real axis.) Functionf generates a continuous functional on L1(0, 1) by means of the formula(f, x) =Z 10 f(t)x(t)dt.Since F is continuously embedded into L1(0, 1) then f generates a continuous func-tional on F. Let us denote by U the subset of F ∗consisting of all functionals of suchtype.

Let M=clU. It is clear that M is a closed subspace of F ∗.Every element of F may be considered as a functional on M. The correspondingembedding of F into M∗we denote by H.Remark 1.

The subspace M is a total subspace of F ∗. It follows from the followingfacts: 1) F is injectively embedded into L1(0, 1); 2) the set of functionals generated bypolynomials is total in (L1(0, 1))∗.The main result of the present paper is the followingTHEOREM 1.

Let cl(H(F)) be of infinite codimension in M∗and let G containsa linearly independent infinite sequence of functions which have analytic continuationsonto some open region ∆of the complex plane such that ∆⊃[0, 1]. Then for everycountable ordinal α there exists a linear continuous injective integral operator T : F →G with analytic kernel such that T −1 : TF →F does not belong to Φα.Let us introduce necessary definitions.

Let X be a Banach space. Weak∗sequentialclosure of a subset V of X∗is defined to be the set of all limits of weak∗convergentsequences from V .

Weak∗sequential closure will be denoted by V(1). For ordinal αweak∗sequential closure of order α of a subset V of X∗is defined by the equalityV(α) = ∪β<α(V(β))(1).THEOREM 2 (A.N.Plichko [10]).

Let X be a separable Banach space and Y bean arbitrary Banach space. Let T : X →Y be a continuous linear injective operator.Let T −1 : TX →X belongs to Φα.

If α is finite then (T ∗Y ∗)(α) = X∗. If α is infinitethen (T ∗Y ∗)(α+2) = X∗.

Conversely, if α is finite and (T ∗Y ∗)(α) = X∗or if α is infiniteand (T ∗Y ∗)(α+1) = X∗, then the mapping T −1 : TX →X belongs to Φα.The proof of Theorem 1 is based on Theorem 2 and the following proposition.Proposition. Let F and M satisfy the assumptions of Theorem 1.

Then for everycountable ordinal β there exists a subspace K of M such that K(β) ̸= F ∗.Proof.Using the result due to Davis and Johnson [2, p. 360] we find a weak∗null sequence {un}∞n=1 in M and a bounded sequence {vk}∞k=1 in F ∗∗such that for somepartition {Ik}∞k=1 of the set of natural numbers onto the pairwise disjoint infinite subsets2

we shall havevk(un) = 1,if n ∈Ik;0,if n ̸∈Ik.Using the arguments from the proof of Theorem III.1 in [3] we select a weak∗basicsubsequence {un(i)}∞i=1 ⊂{un}∞n=1 such that the intersection Ik ∩{n(i)}∞i=1 is infinitefor every k ∈N. Passing if necessary to a subsequence and renumbering {un(i)}∞i=1 weobtain the sequence {yj}∞j=0 such thatvk(yj) = 1,if j can be represented in the form j = n(n + 1)/2 + k with k ≤n;0,if it is not the case.Definition of the weak∗basic sequence implies that the space Z = F/({yj}∞j=0)⊤hasthe basis {zj}∞j=0 such that yj(zk) = δj,k.

It is easy to see that the sequence {zj}∞j=0 isbounded away from zero and that the set {Pki=j zi(i+1)/2+j}∞∞j=0,k=j is bounded.We may assume without loss of generality that ||zi|| ≤1 for every i ∈N. Using thearguments from the proof of lemma 1 in [7] we find a subspace N of cl(lin{yj}∞j=0) anda bounded sequence {hn}∞n=1 in Z∗∗such that the following conditions are satisfied:a) If a weak∗convergent sequence {x∗m}∞m=1 is contained in N(γ) for some γ < β andx∗= w∗−limm→∞x∗m thenhn(x∗) = limm→∞hn(x∗m)for every n ∈N.b) There exists a collection {x∗n,m}∞∞n=1,m=1 of vectors of N(β) such that for everyk, n ∈N we havew∗−limm→∞x∗n,m = 0;(∀m ∈N)(h∗k(x∗n,m) = δk,n).Let {s∗k}∞k=1 be a total (over F) sequence in M. Let c1 = supn ||hn||.

Let νn > 0 (n ∈N)be such that P∞n=1 νn < 1/(2c1).We shall identify Z∗with its natural image in F ∗. Without loss of generality we mayassume that Z∗̸= F ∗.Let us denote by gn(n ∈N) norm-preserving extensions of hn onto F ∗.Let usintroduce the operator R : F ∗→F ∗by the equalityR(x∗) = x∗+∞Xn=1νngn(x∗)s∗n.It is easy to check that ||R −I|| ≤1/2 (where I is the identity operator), hence R isan isomorphism.Let K = R(N).

In the same way as in [7] we prove that for every γ ≤β we haveK(γ) = R(N(γ)). (1)Since R is an isomorphism then relations K(β) = R(N(β)) and N(β) ⊂Z∗̸= F ∗implythat K(β) ̸= F ∗.

At the same time (1) and b) imply that R(x∗n,m) = x∗n,m + νns∗n ∈K(β)and therefore s∗n ∈K(β+1). By the totality of {s∗n} it follows that the subspace K ⊂F ∗is total.

The proof of the proposition is complete.3

Proof of Theorem 1. Using the proposition for β = α + 2 we find a subspace Kof M such that K(α+2) ̸= F ∗.

Using well-known arguments (see e.g. [5, p. 43,44]) wefind a fundamental minimal sequence {fi}∞i=1 in the space F such that its biorthogonalfunctionals {f ∗i }∞i=1 are total and are contained in K. It is easy to see (see e.g.

[8])that there exists an isomorphism S : F →F such that functionals S∗f ∗i (i ∈N) arecontained in U. Let us introduce the notation g∗i = S∗f ∗i (i ∈N).

It is clear that(cl(lin({g∗i }∞i=1)))(α+2) ̸= F ∗. (2)Reducing if necessary the domain ∆we may assume that ∆is bounded and that Gcontains a linearly independent sequence of functions which have analytic continuationsonto ∆which are continuous on the closure of ∆.

Using standard biorthogonalizationprocedure (see e.g. [5, p. 43,44]) we find a minimal sequence {qi}∞i=1 in G such thatthere exist analytic continuations {˜qi}∞i=1 of {qi}∞i=1 to ∆which are continuous on theclosure of ∆.Let us introduce numbersai = max{||qi||G, supt∈∆|˜qi(t)|}, (i ∈N).Let ri(i ∈N) be functions analytic in Γ and continuous on the closure of Γ such thatfunctionals generated by them on F coincide with g∗i (i ∈N).

Let us introduce thenumbersbi = max{||g∗i ||F∗, supt∈Γ |ri(t)|}, (i ∈N).Let us introduce the operator T : F →G by the equalityT(f) =∞Xi=1g∗i (f)qi/(2iaibi). (3)This operator may be considered as an integral operator with the kernelK(t1, t2) =∞Xi=1ri(t1)˜qi(t2)/(2iaibi).By the definitions of ai and bi it follows that this series converges to a function analyticin Γ × ∆.Operator T is injective since the sequence {g∗i }∞i=1 is total and the sequence {qi}∞i=1 isminimal.

It is easy to see that T ∗G∗⊂cl(lin({g∗i }∞i=1)). By (2) we obtain (T ∗G∗)(α+2) ̸=F ∗.

By Theorem 2 it follows that T −1 does not belong to Φα. The theorem is proved.Remark 2.

It is easy to see that if F and G are infinite dimensional function spacesand T : F →G is an injective integral operator with an analytic kernel then G satisfiesthe condition of Theorem 1.Remark 3.If the subspace M ⊂F ∗is norming then the infinite codimensionof cl(H(F)) in M∗is equivalent to the infinite codimension of lin(M⊥∪F) in F ∗∗.Therefore Theorems 1 and 2 of [9] may be obtained using the arguments of the presentpaper. Furthermore, it follows that many of the spaces considered in [9] satisfy the4

condition of Theorem 1 of the present paper. In particular it is the case for classicalnonreflexive spaces.Remark 4.

Let F be a nonseparable function space, which is continuously injec-tively embedded into L1(0, 1), and G is the space satisfying the condition of Theorem 1.Let {g∗i }∞i=1 be a total sequence in F ∗such that functionals g∗i (i ∈N) are represented byfunctions analytic in the same open subset containing [0,1]. Let {qi}∞i=1 be the sequenceintroduced in the proof of Theorem 1.

Let us introduce the operator T : F →G by (3).The operator T is an injective integral operator with an analytic kernel. It is clear thatthe image of T is separable.

Therefore the mapping T −1 : TF →F is not analyticallyrepresentable since the image of the separable space under the action of analyticallyrepresentable mapping is separable.REFERENCES1. Banach S. ¨Uber analytisch darstellbare Operationen in abstrakten Raumen, Fund.Math.

17 (1931), 283–295.2. Davis W.J., Johnson W.B.

Basic sequences and norming subspaces in non-quasi-reflexive Banach spaces, Israel J. Math.

14 (1973), 353–367.3. Johnson W.B., Rosenthal H.P.

On w∗-basic sequences and their applications tothe study of Banach spaces, Studia Math. 43 (1972), 77–92.4.

Kuratowski K. Topology, v. I, New York, Academic Press, 1966.5. Lindenstrauss J., Tzafriri L. Classical Banach spaces, v. I, Berlin, Springer-Verlag,1977.6.

Menikhes L.D. On the regularizability of mappings inverse to integral operators,Doklady AN SSSR, 241 (1978), no.

2, 282–285 (Russian).7.Ostrovskii M.I. Total subspaces with long chains of nowhere norming weak∗sequential closures, Note Mat., to appear.8.

Ostrovskii M.I. Regularizability of inverse linear operators in Banach spaces withbases, Sibirsk.

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33 (1992), no. 3, 123–130 (Russian).9.

Plichko A.N. Non-norming subspaces and integral operators with non-regulari-zable inverses, Sibirsk.

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29 (1988), no. 4, 208–211 (Russian).10.

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Rolewicz S. On the inversion of non-linear transformations, Studia Math. 17(1958), 79–83.12.

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