2022/01/31 by Paolo Leonetti, Leonetti, Paolo · 2 citations
Mathematics · #Advanced Topics in Algebra #Approximation Theory and Sequence Spaces #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory
paper · pdf · doi:10.48550/arxiv.2201.13059
openalex publication_date 2022/01/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let X,Y be Banach spaces, and fix a linear operator T ∈ L(X,Y), and ideals I, J on ω. We obtain Silverman--Toeplitz type theorems on matrices A=(An,k: n,k ∈ ω) of linear operators in L(X,Y), so that J-lim Ax=T(\hspace.2mmI-lim x) for every X-valued sequence x=(x0,x1,…) which is I-convergent [and bounded]. This allows us to establish the relationship between the classical Silverman--Toeplitz characterization of regular matrices and its multidimensional analogue for double sequences, its variant for matrices of linear operators, and the recent version (for the scalar case) in the context of ideal convergence. As byproducts, we obtain characterizations of several matrix classes and a generalization of the classical Hahn--Schur theorem. In the proofs we will use an ideal version of the Banach--Steinhaus theorem which has been recently obtained by De Bondt and Vernaeve in [J.~Math.~Anal.~Appl.~495 (2021)].