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On the growth of resolvent of Toeplitz operators

Leonid Golinskii, Stanislas Kupin, Anna Vishnyakova

Abstract

We study the growth of the resolvent of a Toeplitz operator $T_b$, defined on the Hardy space, in terms of the distance to its spectrum $σ(T_b)$. We are primarily interested in the case when the symbol $b$ is a Laurent polynomial (\emph{i.e., } the matrix $T_b$ is banded). We show that for an arbitrary such symbol the growth of the resolvent is quadratic, and under certain additional assumption it is linear. We also prove the quadratic growth of the resolvent for a certain class of non-rational symbols.

On the growth of resolvent of Toeplitz operators

Abstract

We study the growth of the resolvent of a Toeplitz operator , defined on the Hardy space, in terms of the distance to its spectrum . We are primarily interested in the case when the symbol is a Laurent polynomial (\emph{i.e., } the matrix is banded). We show that for an arbitrary such symbol the growth of the resolvent is quadratic, and under certain additional assumption it is linear. We also prove the quadratic growth of the resolvent for a certain class of non-rational symbols.
Paper Structure (5 sections, 6 theorems, 96 equations)

This paper contains 5 sections, 6 theorems, 96 equations.

Table of Contents

  1. Jordan property and regular Laurent polynomials
  2. Laurent polynomials with Jordan property
  3. Regular Laurent polynomials
  4. Linear growth of the resolvent
  5. Quadratic growth of the resolvent

Key Result

Proposition 1.2

Assume that $\max(m,k)\ge2$, and one of the two conditions holds Then $b({\mathbb{T}})$ is a Jordan curve. Moreover, if one of the strict inequalities holds then $b$ is a LJ-polynomial.

Theorems & Definitions (17)

  • Definition 1.1
  • Proposition 1.2
  • proof
  • Example 1.3
  • Lemma 1.4
  • proof
  • Definition 1.5
  • Remark 1.6
  • Theorem 1.7
  • proof
  • ...and 7 more