Convergence of Complementable Operators

TL;DR

This paper investigates the convergence properties and topological structure of complementable operators in infinite-dimensional Hilbert spaces, extending classical matrix decompositions and providing new insights into their behavior.

Contribution

It offers new results on the convergence of sequences of complementable operators and characterizes the topological structure of their set in the strong operator topology.

Findings

Limit of complementable operators can remain complementable under certain conditions.

Sequences of powers of complementable operators have specific convergence behaviors.

The set of complementable operators forms a boundary subset in the strong operator topology.

Abstract

Complementable operators extend classical matrix decompositions, such as the Schur complement, to the setting of infinite-dimensional Hilbert spaces, thereby broadening their applicability in various mathematical and physical contexts. This paper focuses on the convergence properties of complementable operators, investigating when the limit of sequence of complementable operators remains complementable. We also explore the convergence of sequences and series of powers of complementable operators, providing new insights into their convergence behavior. Additionally, we examine the conditions under which the set of complementable operators is the subset of set of boundary points of the set of non-complementable operators with respect to the strong operator topology. The paper further explores the topological structure of the subset of complementable operators, offering a characterization of its closed subsets.