Rigidity of weighted composition operators on $H^p$

Abstract

We show that every non-compact weighted composition operator $f\mapsto u\cdot (f\circ \varphi )$ acting on a Hardy space $H^p$ for $1\le p<\infty$ 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.

4 pages