Partially isometric Toeplitz operators on the polydisc

TL;DR

This paper characterizes partially isometric Toeplitz operators on the polydisc, linking their structure to inner functions, and extends classical results with new proofs and insights into their properties like hyponormality and power partial isometry.

Contribution

It provides a complete characterization of partially isometric Toeplitz operators on the polydisc, including new proofs of classical theorems and analysis of their spectral and structural properties.

Findings

Partially isometric Toeplitz operators are characterized by inner functions depending on different variables.

Such operators are hyponormal if and only if their symbols are inner functions.

They are always power partial isometries, classified as shifts, co-shifts, or sums of truncated shifts.

Abstract

A Toeplitz operator $T_\varphi$, $\varphi \in L^\infty(\mathbb{T}^n)$, is a partial isometry if and only if there exist inner functions $\varphi_1, \varphi_2 \in H^\infty(\mathbb{D}^n)$ such that $\varphi_1$ and $\varphi_2$ depends on different variables and $\varphi = \bar{\varphi}_1 \varphi_2$. In particular, for $n=1$, along with new proof, this recovers a classical theorem of Brown and Douglas. \noindent We also prove that a partially isometric Toeplitz operator is hyponormal if and only if the corresponding symbol is an inner function in $H^\infty(\mathbb{D}^n)$. Moreover, partially isometric Toeplitz operators are always power partial isometry (following Halmos and Wallen), and hence, up to unitary equivalence, a partially isometric Toeplitz operator with symbol in $L^\infty(\mathbb{T}^n)$, $n > 1$, is either a shift, or a co-shift, or a direct sum of truncated shifts. Along the way, we prove that $T_\varphi$ is a shift whenever $\varphi$ is inner in $H^\infty(\mathbb{D}^n)$.