On contractive and convergence properties of a class of truncated operators represented through Schauder bases with applications
Let \((X,d)\) be a separable complete linear metric space and let \(\{e_n\}_{n=1}^{\infty}\subset X\) be a Schauder basis, that is, the representation \[ x=\sum_{n=1}^{\infty} x_i e_i, \] is unique, for all \(x\in X\). By considering the truncated operators represented through the Schauder basis \(\{e_n\}_{n=1}^{\infty}\subset X\), \(P_m: X\rightarrow X_m\), defined by \[ [x]_m:=P_m x=\sum_{i=1}^{m} x_i e_i, \] the author investigates the relationships between the fixed points of asymptotically contractive operators and their obtained truncations of the expansions of such operators by using Schauder bases. The problem is considered on complete linear metric spaces, in general, and in separable Hilbert spaces in particular. Two illustrative worked examples are given.
homogeneous and translation-invariant metric
scientific article; zbMATH DE number 6418466
scientific article