On Schauder’s Theorem and $s$-numbers (Daniel Akech Thiong, CGU)

Let \(\mathcal{L}(X,Y)\) denote the normed vector space of all continuous operators from \(X\) to \(Y\), \(X^*\) be the dual space of \(X\), and \(\mathcal{K}(X,Y)\) denote the collection of all compact operators from \(X\) to \(Y\). Denote by \(T^{*} \in \mathcal{L}(Y^{*}, X^{*} )\) the adjoint operator of \(T\in \mathcal{L} (X, Y)\). The well known theorem of Schauder states that \(T \in \mathcal{K}(X,Y) \iff T^{*} \in \mathcal{K}(Y^{*},X^{*})\). When an operator fails to be compact, it is sometimes useful to be able to quantify the degree to which it fails to be compact, which has led to the introduction of certain approximation quantities, usually called \(s\)-numbers, and are closely related to singular values. Specifically, the concept of \(s\)-numbers, \(s_n(T)\), arises from the need to assign to every operator \(T: X \to Y\) a certain sequence of numbers \(\{s_n(T)\}\) such that \[s_1(T) \geq s_2(T) \geq \dots \geq 0\] which characterizes the degree of compactness/non-compactness of \(T\). The main examples of \(s\)-numbers include approximation numbers and Kolmogorov numbers. Motivated by Schauder’s theorem, in this talk I will present the relationship between various \(s\)-numbers of an operator \(T\) and its adjoint \(T^*\) between Banach spaces. Joint work with Asuman G. Aksoy.

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