Borel properties of linear operators

Given an injective bounded linear operator T :X→Y between Banach spaces, we study the Borel measurability of the inverse map T −1 :TX→X. A remarkable result of Saint-Raymond (Ann. Inst. Fourier (Grenoble) 26 (1976) 211–256) states that if X is separable, then the Borel class of T −1 is α if, and only if, X∗ is the αth iterated sequential weak∗-closure of T ∗Y∗ for some countable ordinal α. We show that Saint-Raymond’s result holds with minor changes for arbitrary Banach spaces if we assume that T has certain property named co-σ -discreteness after Hansell (Proc. London Math. Soc. 28 (1974) 683–699). As an application, we show that the Borel class of the inverse of a co-σ -discrete operator T can be estimated by the image of the unit ball or the restrictions of T to separable subspaces of X. Our results apply naturally when X is a WCD Banach space since in this case any injective bounded linear operator defined on X is automatically co-σ -discrete.  2003 Elsevier Inc. All rights reserved.