Universal composition operators

TL;DR

This paper characterizes universal composition operators on Hardy spaces, revealing new examples and providing insights into the invariant subspace problem through the study of minimal invariant subspaces and eigenvectors.

Contribution

It offers a complete characterization of linear fractional composition operators with universal translates on Hardy spaces, including a novel example with affine symbol.

Findings

Characterization of all linear fractional composition operators with universal translates.

Identification of a new example of a universal composition operator with affine symbol.

Insights into the invariant subspace problem via minimal invariant subspaces and eigenvectors.

Abstract

A Hilbert space operator $U$ is called universal (in the sense of Rota) if every Hilbert space operator is similar to a multiple of $U$ restricted to one of its invariant subspaces. It follows that the Invariant Subspace Problem for Hilbert spaces is equivalent to the statement that all minimal invariant subspaces for $U$ are one dimensional. In this article we characterize all linear fractional composition operators $C_{\phi} f=f\circ\phi$ that have universal translates on both the classical Hardy spaces $H^2(\mathbb{C}_+)$ and $H^2(\mathbb{D})$ of the half-plane and the unit disk respectively. The surprising new example is the composition operator on $H^2(\mathbb{D})$ with affine symbol $\phi_a(z)=az+(1-a)$ for $0<a<1$. This leads to strong characterizations of minimal invariant subspaces and eigenvectors of $C_{\phi_a}$ and offers an alternative approach to the ISP.