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Which hyponormal block Toeplitz operators are either normal or analytic?

Senhua Zhu, Yufeng Lu, Chao Zu

TL;DR

This work extends the Curto–Hwang–Lee analysis of hyponormal versus subnormal behavior to block Toeplitz operators with matrix-valued symbols of bounded type. By leveraging Douglas–Shapiro–Shields factorization, gcd/lcm theory for matrix-valued inner functions, and the minimal scalar inner multiple $D(\Theta)$, the authors derive conditions under which a hyponormal $T_{\Phi}$ with $T_{\Phi}^{2}$ hyponormal must be normal or analytic. A key contribution is solving CHL’s Problem 10.6 in a broad setting, showing that hyponormality plus bounded-type symbols and a scalar inner inner-part force normality or analyticity when appropriate coprimality holds. The results deepen the link between operator-theoretic hyponormality/subnormality and the function-theoretic structure of matrix-valued inner functions, with implications for multivariable operator theory and mathematical physics.

Abstract

In this paper, we continue Curto-Hwang-Lee's work to study the connection between hyponormality and subnormality for block Toeplitz operators acting on the vector-valued Hardy space of the unit circle. Curto-Hwang-Lee's work focuses primarily on hyponormality and subnormality of block Toeplitz operators with rational symbols. By studying the greatest common divisor of matrix-valued inner functions and the ``weak" commutativity of matrix-valued inner functions, we extended Curto-Hwang-Lee's result to block Toeplitz operators with symbols of bounded type. More precisely, we proved that if $Ψ,Ψ^{\ast}$ are matrix-valued functions of bounded type and the inner part of $Ψ$ of Douglas-Shapiro-Shields factorization is a scalar inner function, then every hyponormal Toeplitz operator $T_Ψ$ whose square is also hyponormal must be either normal or analytic.

Which hyponormal block Toeplitz operators are either normal or analytic?

TL;DR

This work extends the Curto–Hwang–Lee analysis of hyponormal versus subnormal behavior to block Toeplitz operators with matrix-valued symbols of bounded type. By leveraging Douglas–Shapiro–Shields factorization, gcd/lcm theory for matrix-valued inner functions, and the minimal scalar inner multiple , the authors derive conditions under which a hyponormal with hyponormal must be normal or analytic. A key contribution is solving CHL’s Problem 10.6 in a broad setting, showing that hyponormality plus bounded-type symbols and a scalar inner inner-part force normality or analyticity when appropriate coprimality holds. The results deepen the link between operator-theoretic hyponormality/subnormality and the function-theoretic structure of matrix-valued inner functions, with implications for multivariable operator theory and mathematical physics.

Abstract

In this paper, we continue Curto-Hwang-Lee's work to study the connection between hyponormality and subnormality for block Toeplitz operators acting on the vector-valued Hardy space of the unit circle. Curto-Hwang-Lee's work focuses primarily on hyponormality and subnormality of block Toeplitz operators with rational symbols. By studying the greatest common divisor of matrix-valued inner functions and the ``weak" commutativity of matrix-valued inner functions, we extended Curto-Hwang-Lee's result to block Toeplitz operators with symbols of bounded type. More precisely, we proved that if are matrix-valued functions of bounded type and the inner part of of Douglas-Shapiro-Shields factorization is a scalar inner function, then every hyponormal Toeplitz operator whose square is also hyponormal must be either normal or analytic.
Paper Structure (11 sections, 25 theorems, 176 equations)

This paper contains 11 sections, 25 theorems, 176 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Factorization of matrix-valued inner functions
  4. Matrix-valued functions of bounded type
  5. The minimal scalar inner multiple $D(\Theta)$
  6. Two-sided inner functions
  7. The least common inner multiple of two-sided inner functions
  8. The greatest common inner divisor of two-sided inner functions
  9. The proof of Theorem \ref{['shly10']}
  10. "Weak" commutativity of two-sided inner functions
  11. Subnormality of block Toeplitz operators

Key Result

Lemma 2.1

Let $\Theta_{i}=\theta_{i}I_{n}$ for an scalar inner function $\theta_{i}, i\in J$. Let $\theta_{d}=\text{g.c.d.}\{\theta_{i}:i \in J\}$ and $\theta_{m}=\text{l.c.m.}\{\theta_{i}:i \in J\}$, then

Theorems & Definitions (56)

  • Remark 2.1
  • Lemma 2.1: Lemma 2.1 in CHL2012
  • Lemma 2.2: Lemma 2.2 in CHL2021
  • Definition 2.1
  • Lemma 2.3: Theorem 4.13 in CHL2019, Lemma C.13 in CHL2021
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • ...and 46 more