Equivalence after extension and matricial coupling coincide with Schur coupling, on separable Hilbert spaces

TL;DR

This paper proves that on separable Hilbert spaces, operators that are equivalent after extension or matricially coupled are also Schur coupled, extending previous results beyond Fredholm operators.

Contribution

It generalizes the equivalence of Schur coupling and extension/matricial coupling to all operators on separable Hilbert spaces.

Findings

Equivalence after extension implies Schur coupling for operators with closed range.

Operators approximable by invertible operators also satisfy this implication.

The result now holds for all operators on separable Hilbert spaces.

Abstract

It is known that two Banach space operators that are Schur coupled are also equivalent after extension, or equivalently, matricially coupled. The converse implication, that operators which are equivalent after extension or matricially coupled are also Schur coupled, was only known for Fredholm Hilbert space operators and Fredholm Banach space operators with index 0. We prove that this implication also holds for Hilbert space operators with closed range, generalizing the result for Fredholm operators, and Banach space operators that can be approximated in operator norm by invertible operators. The combination of these two results enables us to prove that the implication holds for all operators on separable Hilbert spaces.