Faculty Publications

The Chalmers-Metcalf Operator And Minimal Extensions

Document Type

Article

Keywords

Minimal extension, Minimal projection

Journal/Book/Conference Title

Journal of Functional Analysis

Volume

280

Issue

1

Abstract

Let X be a real or complex Banach space and let Y⊂X be a finite-dimensional subspace. Fix A∈L(Y). Then we define PA(X,Y)={P∈L(X,Y):P|Y=A}. An operator Po∈PA(X,Y) is called a minimal extension (a minimal projection if A=Y) if ‖Po‖=inf⁡{‖P‖:P∈PA(X,Y)}. The aim of this paper is to present a variety of theorems characterizing minimal extensions, which generalize previously obtained results (in the real case) for minimal projections. We include several new applications, in which these theorems are utilized to determine minimal projections. Moreover, these characterizations employ so-called Chalmers-Metcalf operators (which are defined within the context of Theorem 1) and the form of these operators (when properly restricted) is also considered here. Indeed, we show that under certain assumptions, this form becomes quite simple - essentially the identity map - and this is of benefit in determining minimal extensions. We note that it has been conjectured that the assumptions we put in place to guarantee this simple form can be significantly weakened and we address this question.

Department

Department of Mathematics

Original Publication Date

1-1-2021

DOI of published version

10.1016/j.jfa.2020.108800

Repository

UNI ScholarWorks, Rod Library, University of Northern Iowa

Language

en

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