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Conjugations of Unitary Operators, II

Javad Mashreghi, Marek Ptak, William T. Ross

Abstract

For a given unitary operator $U$ on a separable complex Hilbert space $\h$, we describe the set $\mathscr{C}_{c}(U)$ of all conjugations $C$ (antilinear, isometric, and involutive maps) on $\h$ for which $C U C = U$. As this set might be empty, we also show that $\mathscr{C}_{c}(U) \not = \varnothing$ if and only if $U$ is unitarily equivalent to $U^{*}$.

Conjugations of Unitary Operators, II

Abstract

For a given unitary operator on a separable complex Hilbert space , we describe the set of all conjugations (antilinear, isometric, and involutive maps) on for which . As this set might be empty, we also show that if and only if is unitarily equivalent to .
Paper Structure (10 sections, 32 theorems, 144 equations)

This paper contains 10 sections, 32 theorems, 144 equations.

Table of Contents

  1. Introduction
  2. Basics facts about conjugations
  3. Commuting conjugations of unitary matrices
  4. Conjugations and spectral measures
  5. Natural conjugations on vector valued $L^2$ spaces
  6. Conjugations and bilateral shifts
  7. Unitary multiplication operators on $L^2(m, \mathbb{T})$
  8. Conjugations via the spectral theorem
  9. Examples
  10. A remark about invariant subspaces

Key Result

Theorem 1.2

For a unitary operator $U$ on a complex separable Hilbert space $\mathcal{H}$, the following are equivalent.

Theorems & Definitions (65)

  • Theorem 1.2
  • Example 2.2
  • Lemma 2.3
  • Example 2.4
  • Proposition 2.6
  • Lemma 2.7
  • Proposition 2.8
  • Lemma 2.9
  • proof
  • Proposition 3.1
  • ...and 55 more