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Binormal block Toeplitz operators with matrix valued circulant symbols

Nihat Gokhan Gogus, Rewayat Khan, Eungil Ko, Ji Eun Lee

TL;DR

The paper addresses binormality of block Toeplitz operators with matrix-valued circulant symbols, and investigates Γ-dilations and invariant subspaces. By exploiting circulant diagonalization, it reduces binormality questions to scalar diagonal blocks $T_{\lambda_k}$ and establishes unitary equivalences to diagonal Toeplitz operators. It introduces a ${\bf{\Gamma}}$-dilation that preserves diagonal block structure, showing that dilated symbols yield reducing subspaces. It also provides explicit binormal and normal criteria for products of four commuting normal matrix-valued symbols and demonstrates several illustrative examples.

Abstract

This paper focuses on the binormality of block Toeplitz operators with matrix valued circulant symbols. We also study some Γ-dilations of Toeplitz operators. Moreover, we also analyze the invariant subspace of Toeplitz operators with matrix-valued symbols.

Binormal block Toeplitz operators with matrix valued circulant symbols

TL;DR

The paper addresses binormality of block Toeplitz operators with matrix-valued circulant symbols, and investigates Γ-dilations and invariant subspaces. By exploiting circulant diagonalization, it reduces binormality questions to scalar diagonal blocks and establishes unitary equivalences to diagonal Toeplitz operators. It introduces a -dilation that preserves diagonal block structure, showing that dilated symbols yield reducing subspaces. It also provides explicit binormal and normal criteria for products of four commuting normal matrix-valued symbols and demonstrates several illustrative examples.

Abstract

This paper focuses on the binormality of block Toeplitz operators with matrix valued circulant symbols. We also study some Γ-dilations of Toeplitz operators. Moreover, we also analyze the invariant subspace of Toeplitz operators with matrix-valued symbols.

Paper Structure

This paper contains 5 sections, 18 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Binormal block Toeplitz operators with matrix valued circulant symbols
  4. $\Gamma$-dilation of Toeplitz operators
  5. Binormal Toeplitz operators with matrix valued symbols

Key Result

Theorem 1.1

FM Let $T\in\mathcal{L}(E)$ and let $\mathcal{M}$ be a non-trivial closed subspace of $E$. Then the matrix representation of $T$ with respect to the decomposition $E=\mathcal{M}\oplus \mathcal{M}^{\perp}$ is block diagonal if and only if the subspace $\mathcal{M}$ is reducing for $T$.

Theorems & Definitions (36)

  • Theorem 1.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • proof
  • Theorem 3.5
  • proof
  • Corollary 3.6
  • ...and 26 more