Representation of Generalized Bi-Circular Projections on Banach Spaces

TL;DR

This paper investigates the structure of generalized bi-circular projections on Banach spaces, providing new representations, examples, and characterizations, especially on spaces like $C_0( ext{Om},X)$ with specific properties.

Contribution

It introduces novel results on representing projections on Banach spaces, including an example of a generalized bi-circular projection that cannot be expressed as an average of the identity and an isometric reflection.

Findings

Characterization of generalized bi-circular projections on $C_0( ext{Om},X)$

Existence of generalized bi-circular projections not expressible as averages of identity and reflection

New representation results for projections on arbitrary Banach spaces

Abstract

We prove several results concerning the representation of projections on arbitrary Banach spaces. We also give illustrative examples including an example of a generalized bi-circular projection which can not be written as the average of the identity with an isometric reflection. We also characterize generalized bi-circular projections on $C_0(\Om,X)$, with $\Om$ a locally compact Hausdorff space (not necessarily connected) and $X$ a Banach space with trivial centralizer.