On the connected components of the conjugacy class of projectors on $ \ell_p\oplus\ell_q $

TL;DR

This paper characterizes projectors on Banach spaces that are connected via conjugation, showing examples where conjugacy classes split into multiple components and describing classes on certain direct sum spaces.

Contribution

It provides a new characterization of conjugacy-connected projectors and analyzes the structure of their conjugacy classes on specific Banach spaces.

Findings

Conjugacy classes can split into multiple path-connected components.

Characterization of projectors connected via conjugation.

Description of conjugacy classes on lp_q lp_q spaces.

Abstract

We characterize the projectors $ P $ on a Banach space $ E $ having the property of being connected to all the others projectors obtained as a conjugation of $ P $. Using this characterization we show an example of Banach space where the conjugacy class of a projector splits into several path-connected components, and describe the conjugacy classes of projectors onto subspaces of $ \ell_p\oplus\ell_q $ with $ p\neq q $.