On the closedness of the sum of ranges of operators $A_k$ with almost compact products $A_i^* A_j$

TL;DR

This paper investigates the conditions under which the sum of the ranges of operators with almost compact products remains closed, extending known results from strictly compact to nearly compact cases in Hilbert spaces.

Contribution

It introduces the concept of 'almost' compact products and proves that the sum of the ranges is closed under these near-compact conditions, generalizing previous theorems.

Findings

Sum of ranges is closed when products are almost compact.

Ranges are essentially linearly independent under these conditions.

Extends classical results from compact to almost compact operators.

Abstract

Let $\mathcal{H}_1,\ldots,\mathcal{H}_n,\mathcal{H}$ be complex Hilbert spaces and $A_k:\mathcal{H}_k\to\mathcal{H}$ be a bounded linear operator with the closed range $Ran(A_k)$, $k=1,\ldots,n$. It is known that if $A_i^*A_j$ is compact for any $i\neq j$, then $\sum_{k=1}^n Ran(A_k)$ is closed. We show that if all products $A_i^*A_j$, $i\neq j$ are "almost" compact, then the subspaces $Ran(A_1),\ldots,Ran(A_n)$ are essentially linearly independent and their sum is closed.