Matrix splitting and ideals in $B(H)$

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.