Borel properties of linear operators

Given an injective bounded linear operator <f>T :X→Y</f> between Banach spaces, we study the Borel measurability of the inverse map <f>T−1 :TX→X</f>. A remarkable result of Saint-Raymond (Ann. Inst. Fourier (Grenoble) 26 (1976) 211–256) states that if <f>X</f> is separable, then the Borel class of <f>T−1</f> is <f>α</f> if, and only if, <f>X&ast;</f> is the <f>α</f>th iterated sequential weak<f>&ast;</f>-closure of <f>T&ast;Y&ast;</f> for some countable ordinal <f>α</f>. We show that Saint-Raymond''s result holds with minor changes for arbitrary Banach spaces if we assume that <f>T</f> has certain property named co-<f>σ</f>-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-<f>σ</f>-discrete operator <f>T</f> can be estimated by the image of the unit ball or the restrictions of <f>T</f> to separable subspaces of <f>X</f>. Our results apply naturally when <f>X</f> is a WCD Banach space since in this case any injective bounded linear operator defined on <f>X</f> is automatically co-<f>σ</f>-discrete. [Copyright &y& Elsevier]