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The Taylor spectrum of pairs of isometries

Zbigniew Burdak, Patryk Pagacz

Abstract

In the paper we fully describe Taylor spectrum of pairs of isometries given by diagrams. In most cases both isometries in such pairs have non-trivial shift part and its Taylor spectrum is a proper subset (of Lebesgue measure in $(0,π^2)$) of the closed bidisc.

The Taylor spectrum of pairs of isometries

Abstract

In the paper we fully describe Taylor spectrum of pairs of isometries given by diagrams. In most cases both isometries in such pairs have non-trivial shift part and its Taylor spectrum is a proper subset (of Lebesgue measure in ) of the closed bidisc.

Paper Structure

This paper contains 11 sections, 6 theorems, 123 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Spectrum of a single operator
  4. Taylor spectrum of a pair of operators
  5. Pairs of isometries defined by a diagram
  6. Taylor spectrum of pairs of isometries - examples
  7. Parts of Taylor spectrum of the pair of isometries given by diagrams
  8. Condition \ref{['T2']}
  9. Condition \ref{['T3']}
  10. Taylor spectrum of the pair of isometries given by diagrams
  11. Taylor spectrum of pairs of isometries - examples - continued

Key Result

Theorem 3.1

Let $J$ be a diagram equivalent to a subset of $\mathbb{Z}_+^2$ that is $(i,j)+J\subset \mathbb{Z}_+^2$ for some $(i,j)\in\mathbb{Z}^2$. Then where $M_w,M_z\in B(H_J).$ More precisely, $\mathbb{D}^2\subset\Gamma_3(M_w, M_z).$

Theorems & Definitions (33)

  • Remark 2.1
  • proof
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • proof
  • Definition 2.1
  • Remark 2.5
  • proof
  • Remark 2.6
  • ...and 23 more