Inner vectors for Toeplitz operators

TL;DR

This paper surveys various methods for constructing inner functions for Toeplitz operators, extending classical concepts to model spaces and presenting new examples.

Contribution

It consolidates multiple approaches to inner functions for Toeplitz operators and introduces novel examples and extensions to model spaces.

Findings

Unified framework for inner functions in Toeplitz operators

Extension of inner function concepts to model spaces

New examples of inner functions

Abstract

In this paper we survey and bring together several approaches to obtaining inner functions for Toeplitz operators. These approaches include the classical definition, the Wold decomposition, the operator-valued Poisson Integral, and Clark measures. We then extend these notions somewhat to inner functions on model spaces. Along the way we present some novel examples.