Range additivity, shorted operator and the Sherman-Morrison-Woodbury formula

TL;DR

This paper explores the relationship between range additivity, shorted operators, and Hilbert space decompositions, extending a matrix formula to infinite-dimensional operators using these concepts.

Contribution

It establishes connections between range additivity, shorted operators, and compatibility, and generalizes a Sherman-Morrison-Woodbury related formula to infinite-dimensional Hilbert space operators.

Findings

Established relationship between range additivity and shorted operators.

Extended Fill and Fishkind's formula to infinite-dimensional operators.

Provided new insights into operator decompositions in Hilbert spaces.

Abstract

We say that two operators A, B have the range additivity property if R(A + B) = R(A) + R(B). In this article we study the relationship between range additivity, shorted operator and certain Hilbert space decomposition known as compatibility. As an application, we extend to infinite dimensional Hilbert space operators a formula by Fill and Fishkind related to the well-known Sherman-Morrison-Woodbury formula.