On unitary equivalence and reducing subspaces of analytic Toeplitz operator on vector-valued Hardy space

TL;DR

This paper characterizes the unitary equivalence and reducing subspaces of the operator $T_{z^n}$ on vector-valued Hardy spaces, revealing its structure as a direct sum of scalar Toeplitz operators.

Contribution

It proves that $T_{z^n}$ on vector-valued Hardy space is unitarily equivalent to a direct sum of scalar Toeplitz operators and fully describes its reducing subspaces.

Findings

$T_{z^n}$ is unitarily equivalent to $igoplus_1^{mn}T_z$

Complete description of reducing subspaces of $T_{z^n}$

Operator structure elucidated via matrix and operator theory methods

Abstract

In this paper, we proved that $T_{z^n}$ acting on the $\mathbb{C}^m$-valued Hardy space $H_{\mathbb{C}^m}^2(\mathbb{D})$, is unitarily equivalent to $\bigoplus_1^{mn}T_z$, where $T_z$ is acting on the scalar-valued Hardy space $H_{\mathbb{C}}^2(\mathbb{D})$. And using the matrix manipulations combined with operator theory methods, we completely describe the reducing subspaces of $T_{z^n}$ on $H_{\mathbb{C}^m}^2(\mathbb{D})$.