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Spectral cuts and unconventional functional calculi

Eva A. Gallardo-Gutiérrez, F. Javier González-Doña

Abstract

In this work, we prove that linear bounded operators $T$ on a Banach space $X$ allowing spectral cuts along rectifiable Jordan curves meeting their spectrum are related to classes of operators admitting an unconventional functional calculus. We identify several such classes and address the consequences regarding the existence of non-trivial closed invariant subspaces, extending previous results of Chalendar. Furthermore, we establish that every operator belonging to a broad subclass of compact perturbations of diagonalizable normal operators on separable Hilbert spaces, namely, trace-class perturbations, possesses an unconventional functional calculus and is super-decomposable, thereby extending earlier results obtained by the authors.

Spectral cuts and unconventional functional calculi

Abstract

In this work, we prove that linear bounded operators on a Banach space allowing spectral cuts along rectifiable Jordan curves meeting their spectrum are related to classes of operators admitting an unconventional functional calculus. We identify several such classes and address the consequences regarding the existence of non-trivial closed invariant subspaces, extending previous results of Chalendar. Furthermore, we establish that every operator belonging to a broad subclass of compact perturbations of diagonalizable normal operators on separable Hilbert spaces, namely, trace-class perturbations, possesses an unconventional functional calculus and is super-decomposable, thereby extending earlier results obtained by the authors.
Paper Structure (7 sections, 20 theorems, 187 equations, 3 figures)

This paper contains 7 sections, 20 theorems, 187 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Plain spectral cuts and an unconventional functional calculus
  3. Plain spectral cuts along cycles and decomposability
  4. Plain spectral cuts and local resolvent growth
  5. Plain spectral cuts for compact perturbations of normal operators
  6. A first example: plain spectral cuts for normal operators
  7. Trace class perturbations of diagonalizable normal operators

Key Result

Proposition 2.1

Let $T$ be a linear bounded operator on $X$ with the SVEP and $\gamma$ a rectifiable Jordan curve. $T$ admits a plain spectral cut along $\gamma$ if and only if there exists a non-trivial idempotent $P_\gamma \in {\{T\}"}$ such that In such a case, $\sigma(T\mid_{\textnormal{ran}(P_\gamma)})\subseteq \overline{\textnormal{int}(\gamma)}\cap\sigma(T)$ and $\sigma(T\mid_{\ker(P_\gamma)})\subseteq

Figures (3)

  • Figure 1: The set $K=\overline{D}(-1,1)\cup\overline{D}(1,1)$ and a Jordan curve $\gamma$ containing $[-i,i]$ with $\gamma\cap K=\{0\}$.
  • Figure 2: An example of an admissible grid for $T$
  • Figure 3: An appropriate curve for $T$

Theorems & Definitions (54)

  • Definition
  • Proposition 2.1
  • proof
  • Remark 2.2
  • Theorem 2.3
  • proof
  • Remark 2.4
  • Definition 2.5
  • Theorem 2.6
  • proof
  • ...and 44 more