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Product of Matrix Valued Truncated Toeplitz Operators

Muhammad Ahsan Khan

Abstract

Let $A_Φ$ be a matrix valued truncated Toeplitz operator-the compression of multiplication operator to vector-valued model space $H^2(E)\ominus ΘH^2(E)$, where $Θ$ is a matrix valued non constant inner function. Under supplementary assumptions, we find necessary and sufficient condition that the product $A_ΦA_Ψ$ is itself a matrix valued truncated Toeplitz operator.

Product of Matrix Valued Truncated Toeplitz Operators

Abstract

Let be a matrix valued truncated Toeplitz operator-the compression of multiplication operator to vector-valued model space , where is a matrix valued non constant inner function. Under supplementary assumptions, we find necessary and sufficient condition that the product is itself a matrix valued truncated Toeplitz operator.

Paper Structure

This paper contains 5 sections, 8 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Model Spaces and Operators
  3. Truncated Toeplitz Operators and Matrix valued Truncated Toeplitz Operators
  4. Main results
  5. A particular case: Block Toeplitz Matrices

Key Result

Lemma 2.1

Suppose $\Theta(0)=0$, so $\Theta=z\Theta_1$. Let $\Gamma$ be a conjugation on $E$, and suppose that $\Theta(e^{it})^{*}=\Gamma \Theta(e^{it})\Gamma$ a.e. on $\mathbb{T}$. Then the map $C_{\Gamma}$ defined by is a conjugation on $z\mathcal{K}_{\Theta_1}$.

Theorems & Definitions (14)

  • Definition 2.1
  • Lemma 2.1
  • Proposition 3.1
  • Proposition 3.2
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • proof
  • ...and 4 more