Composition operators between Toeplitz kernels

TL;DR

This paper investigates how composition operators affect Toeplitz kernels, identifying conditions for invariance, minimal containing kernels, and explicit maximal vectors, thereby extending understanding of composition operators on model spaces.

Contribution

It determines minimal Toeplitz kernels containing images under composition operators with general inner symbols and extends results to weighted composition operators and model spaces.

Findings

Composition operators preserve nearly $S^*$-invariance only for automorphic inner functions.

Explicit descriptions of maximal vectors for various Toeplitz kernels are provided.

Extensions to weighted composition operators and minimal model spaces are established.

Abstract

Recently, it was shown that the image of a Toeplitz kernel of dimension greater than $1$ under composition by an inner function is nearly $S^*$-invariant if and only if the inner function is an automorphism. Building on this, we determine the minimal Toeplitz kernel containing the image of a Toeplitz kernel under a composition operator with a general inner symbol, and extend this to weighted composition operators. Specifically, the corresponding cases for minimal model spaces are also given, thereby extending known work on the action of composition operators on model spaces. Finally, we use the equivalences between Toeplitz kernels to derive the explicit maximal vectors for several Toeplitz kernels, with symbols expressed in terms of composition operators and inner functions.