# Strict singularity of weighted composition operators on derivative Hardy spaces

## TL;DR

This paper investigates the properties of weighted composition operators on derivative Hardy spaces, establishing a link between strict singularity and compactness, and characterizing operators that fix an isomorphic copy of  for p.

## Contribution

It proves that strict singularity coincides with compactness for these operators and characterizes when they fix an isomorphic copy of  on derivative Hardy spaces.

## Key findings

- Strict singularity of the operator is equivalent to its compactness on S^p.
- Operators fixing an isomorphic copy of  are characterized for p.
- Conditions for weighted composition operators to fix  are provided when p.

## Abstract

We prove that the weighted composition operator $W_{\phi,\varphi}$ fixes an isomorphic copy of $\ell^p$ if the operator $W_{\phi,\varphi}$ is not compact on the derivative Hardy space $S^p$. In particular, this implies that the strict singularity of the operator $W_{\phi,\varphi}$ coincides with the compactness of it on $S^p$. Moreover, when $p\neq2$, we characterize the conditions for those weighted composition operators $W_{\phi,\varphi}$ on $S^p$ which fix an isomorphic copy of $\ell^2$ .