On the closedness of the sum of ranges of operators $A_k$ with almost compact products $A_i^{\ast }{A}_j$

Abstract

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

9 pages

References in corpus (1)

• Topological isomorphisms for some universal operator algebras