Borel properties of linear operators.
The main result of the present paper generalizes a theorem of Saint-Raymond on the Borel classification of the inverse map \(T^{-1}: TX \to X\) of a bounded linear operator \(T: X \to Y\) from a separable Banach space \(X\) to a Banach space \(Y\) to the case of non-separable spaces \(X\). Given \(A \subset X^*\), denote by \(A_{(1)}\) (by \(A_{[1]}\)) the set of the limits of all \(w^*\)-converging sequences (respectively, bounded nets) of elements from \(A\), and define inductively \(A_{(\alpha)}= \bigcup_{\beta < \alpha}(A_{(\beta)})_{(1)}\) (respectively, \(A_{[\alpha]}= \bigcup_{\beta < \alpha}(A_{[\beta]})_{[1]}\)) for any ordinal \(\alpha\). Saint-Raymond's theorem states that if \(X\) is separable and \(T\) is injective then \(T^{-1}\) is of the Borel class \(\alpha\) if and only if \((T^* Y^*)_{(\alpha)} = X^*\). This is not true in general when \(X\) is not separable. To state the main result, recall the following definitions. A family \({\mathcal H} = \{H_i: i \in I \}\) of subsets of a metric space \(X\) is said to be {discrete} if any point \(x \in X\) has a neighborhood \(V\) which meets at most one member of \({\mathcal H}\). The family \({\mathcal H}\) is {\(\sigma\)-discretely decomposable} if there are discrete families \((\{H_i^n: i \in I \})_{n=1}^{\infty}\) such that \(H_i = \bigcup_{n=1}^{\infty} H_i^n\) for every \(i \in I\). Following R.~W.~Hansell, an injective map \(T: X \to Y\) is co-\(\sigma\)-discrete if for every discrete family \(\{H_i: i \in I \}\) in \(X\) the family \(\{T(H_i): i \in I \}\) is \(\sigma\)-discretely decomposable. If \(X\) is separable, then any map is automatically co-\(\sigma\)-discrete. The {bounded-net order} (bn-order) of an \(A \subset X^*\) is defined to be the first ordinal \(\alpha\) with \(A_{[\alpha]} = A_{[\alpha + 1]}\). The main result asserts that for an injective linear bounded co-\(\sigma\)-discrete operator \(T: X \to Y\) between Banach spaces, the inverse \(T^{-1}\) is Borel measurable if and only if the bn-order of \(T^* Y^*\) is countable, and in such a case it coincides with the Borel class of \(T^{-1}\).
scientific article; zbMATH DE number 2055363