Universality arising from invertible weighted composition operators

TL;DR

This paper demonstrates that certain invertible weighted composition operators on Hardy and weighted Bergman spaces exhibit universality, linking their invariant subspace structure to the long-standing Invariant Subspace Problem in Hilbert spaces.

Contribution

It proves the universality of translations by eigenvalues of invertible weighted hyperbolic composition operators on classical function spaces.

Findings

Translations by eigenvalues are universal operators.

Universality relates to the Invariant Subspace Problem.

Results extend to some Banach spaces.

Abstract

A linear operator $U$ acting boundedly on an infinite-dimensional separable complex Hilbert space $H$ is universal if every linear bounded operator acting on $H$ is similar to a scalar multiple of a restriction of $U$ to one of its invariant subspaces. It turns out that characterizing the lattice of closed invariant subspaces of a universal operator is equivalent to solve the Invariant Subspace Problem for Hilbert spaces. In this paper, we consider invertible weighted hyperbolic composition operators and we prove the universality of the translations by eigenvalues of such operators, acting on Hardy and weighted Bergman spaces. Some consequences for the Banach space case are also discussed.