Matrix splitting and ideals in $\mathcal{B}(\mathcal{H})$

TL;DR

This paper explores how the membership of an operator in an ideal relates to its matrix block components, establishing conditions under which these components also belong to the ideal, especially in block band matrix forms.

Contribution

It provides new theorems linking ideal membership of an operator to its matrix block diagonals and their sums, especially in block band matrix representations.

Findings

Operators in an ideal have their block diagonals in the ideal.

Sum of block diagonals lies in the arithmetic mean closure of the ideal.

Principal ideal generated by an operator equals sum of ideals generated by its block diagonals.

Abstract

We investigate the relationship between ideal membership of an operator and its pieces relative to several canonical types of partitions of the entries of its matrix representation with respect to a given orthonormal basis. Our main theorems establish that if $T$ lies in an ideal $\mathcal{I}$, then $\sum P_n T P_n$ (or more generally $\sum Q_n T P_n$) lies in the arithmetic mean closure of $\mathcal{I}$ whenever $\{P_n\}$ (and also $\{Q_n\}$) is a sequence of mutually orthogonal projections; and in any basis for which $T$ is a block band matrix, in particular, when in Patnaik--Petrovic--Weiss universal block tridiagonal form, then all the sub/super/main-block diagonals of $T$ are in $\mathcal{I}$. And in particular, the principal ideal generated by this $T$ is the finite sum of the principal ideals generated by each sub/super/main-block diagonals.