On Analytical Representability of Mappings Inverse to Integral Operators

The condition onto pair (F, G) of function Banach spaces under which there exists a integral operator T : F → G with analytic kernel such that the inverse mapping 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 of mappings from X to Y containing all continuous mappings and being closed with respect to passage to pointwise limit. This family is representable as an union ∪ α∈Ω Φ α , where Ω is the set of all countable ordinals and Φ α are defined in the following way. 1. The class Φ 0 is the set of all continuous mappings. 2. The class Φ α (α > 0) consists of all mappings which are limits of convergent sequences of mappings belonging to ∪ ξ<α Φ ξ. A mapping f : X → Y is called of α Borel class if the set f −1 (F) is a Borel set of multiplicative class α for every closed subset F ⊂Y. It is known [1], [11], if Y is a separable Banach space, then the class Φ α coincides with 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 Φ 1. The present paper is devoted to the following problem. Let F and G be Banach function spaces on [0,1] and T : F → G be an injective integral operator with analytic kernel. What class of analytically representable functions can the mapping T −1 : T F → F belong to? Let us give some clarifications. 1. By analytic kernel we mean the mapping K : [0, 1] × [0, 1] → C with the following property: 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 …