O-frames for operators in Banach spaces (Q2825727)

Let \(X\) be a Banach space. A sequence \((x_k,x_k^*)_{k=1}^\infty\) with \((x_k)_k \subset X\) and \((x_k^*)_k \subset X^*\) is called a Schauder frame for \(X\) if, for every \(x \in X\), the series \(\sum_{k=1}^\infty x_k^*(x) x_k\) converges (in norm) to \(x\).NEWLINENEWLINEIn this paper, the following generalization of Schauder frames is introduced and studied: Let \(T : X \to Y\) be a bounded linear operator between Banach spaces. A sequence \((x_k^*,y_k)_{k=1}^\infty\) with \((x_k^*)_k \subset X^*\) and \((y_k)_k \subset Y\) is an O-frame (operator frame) for \(T\) if, for every \(x \in X\), the series \(\sum_{k=1}^\infty x_k^*(x) y_k\) converges to \(Tx\) in \(Y\). If such a sequence exists, \(T\) is said to have an O-frame.NEWLINENEWLINESeveral examples are given. It is shown that \(T\) has an O-frame if and only if it factors through a Banach (sequence) space with a Schauder basis. If \(X\) is separable, then \(T : X \to Y\) has an O-frame if and only if \(T\) has the bounded approximation property (which means that \(T\) can be approximated uniformly on compact subsets of \(X\) by finite rank operators with norm bounded by a fixed constant times the norm of \(T\)).NEWLINENEWLINEUnconditional O-frames are also studied and characterized. Schauder frames and O-frames are compared and the difference in behavior is pointed out.