On interplay between operators, bases, and matrices

TL;DR

This paper develops methods to construct matrix representations of bounded linear operators on Hilbert spaces with specific algebraic or asymptotic structures, generalizing previous results and answering open questions.

Contribution

It introduces new techniques for creating matrix representations with prescribed diagonals, bounds, and polynomial constraints, expanding on prior work and generalizing classical theorems.

Findings

Constructed matrices with specified diagonal bands

Established bounds for matrix elements

Generalized Stout's theorem and answered an open question

Abstract

Given a bounded linear operator $T$ on separable Hilbert space, we develop an approach allowing one to construct a matrix representation for $T$ having certain specified algebraic or asymptotic structure. We obtain matrix representations for $T$ with preassigned bands of the main diagonals, with an upper bound for all of the matrix elements, and with entrywise polynomial lower and upper bounds for these elements. In particular, we substantially generalize and complement our results on diagonals of operators from [46] and other related results. Moreover, we obtain a vast generalization of a theorem by Stout (1981), and (partially) answer his open question. Several of our results have no analogues in the literature.