Matrix representations of arbitrary bounded operators on Hilbert spaces

TL;DR

This paper demonstrates that, under broad conditions, significant portions of a matrix representing a bounded operator on a Hilbert space can be preassigned, extending previous work mainly focused on diagonals.

Contribution

It generalizes matrix representation results to operator tuples and their powers, providing new variants and broadening the scope of matrix preassignment in infinite-dimensional spaces.

Findings

Large parts of matrices for bounded operators can be preassigned.

Results apply to operator tuples, including powers of a single operator.

Extends previous research on matrix representations in infinite-dimensional spaces.

Abstract

We show that under natural and quite general assumptions, a large part of a matrix for a bounded linear operator on a Hilbert space can be preassigned. The result is obtained in a more general setting of operator tuples leading to interesting consequences, e.g. when the tuple consists of powers of a single operator. We also prove several variants of this result of independent interest. The paper substantially extends former research on matrix representations in infinite-dimensional spaces dealing mainly with prescribing the main diagonals.