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

A.C.M. Ran, S. ter Horst

Research output: Contribution to JournalArticleAcademicpeer-review

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. © 2012 Elsevier Inc. All rights reserved.
Original languageEnglish
Pages (from-to)793-805
JournalLinear Algebra and its Applications
Volume439
DOIs
Publication statusPublished - 2014

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