Rigidity of weighted composition operators on $H^p$

TL;DR

This paper investigates the rigidity properties of weighted composition operators on Hardy spaces, showing they inherently contain copies of sequence spaces and characterizing when they contain Hilbert space copies.

Contribution

It extends previous results by establishing that all non-compact weighted composition operators on $H^p$ contain an isomorphic copy of $ ext{ell}^p$, and characterizes those containing $ ext{ell}^2$.

Findings

Non-compact weighted composition operators fix an isomorphic copy of $ ext{ell}^p$.

Characterization of operators fixing a copy of $ ext{ell}^2$.

Extension of earlier unweighted composition operator results.

Abstract

We show that every non-compact weighted composition operator $f \mapsto u\cdot (f\circ\phi)$ acting on a Hardy space $H^p$ for $1 \leq p < \infty$ fixes an isomorphic copy of the sequence space $\ell^p$ and therefore fails to be strictly singular. We also characterize those weighted composition operators on $H^p$ which fix a copy of the Hilbert space $\ell^2$. These results extend earlier ones obtained for unweighted composition operators.