Hyperinvariant subspace for weighted composition operator on $L^p([0,1]^d)$

TL;DR

This paper proves the existence of hyperinvariant subspaces for weighted composition operators on $L^p([0,1]^d)$ under specific conditions on the weight and the ergodic transformation, extending functional calculus methods.

Contribution

It establishes hyperinvariant subspaces for weighted composition operators with generalized polynomial weights and ergodic transformations, broadening understanding of operator structure on $L^p$ spaces.

Findings

Existence of hyperinvariant subspace under specified conditions

Extension of functional calculus to this class of operators

Application to ergodic transformations with discrepancy conditions

Abstract

The main result of this paper is the existence of a hyperinvariant subspace of weighted composition operator $Tf=vf\circ\tau$ on $L^p([0,1]^d)$, ($1 \leq p \leq \infty$) when the weight $v$ is in the class of ``generalized polynomials'' and the composition map is a bijective ergodic transform satisfying a given discrepancy. The work is based on the construction of a functional calculus initiated by Wermer and generalized by Davie.