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Conjugations of Unitary operators, I

Javad Mashreghi, Marek Ptak, William T. Ross

Abstract

If $U$ is a unitary operator on a separable complex Hilbert space $\mathcal{H}$, an application of the spectral theorem says there is a conjugation $C$ on $\mathcal{H}$ (an antilinear, involutive, isometry on $\mathcal{H}$) for which $ C U C = U^{*}.$ In this paper, we fix a unitary operator $U$ and describe all of the conjugations $C$ which satisfy this property. As a consequence of our results, we show that a subspace is hyperinvariant for $U$ if and only if it is invariant for any conjugation $C$ for which $CUC = U^{*}$.

Conjugations of Unitary operators, I

Abstract

If is a unitary operator on a separable complex Hilbert space , an application of the spectral theorem says there is a conjugation on (an antilinear, involutive, isometry on ) for which In this paper, we fix a unitary operator and describe all of the conjugations which satisfy this property. As a consequence of our results, we show that a subspace is hyperinvariant for if and only if it is invariant for any conjugation for which .
Paper Structure (8 sections, 29 theorems, 113 equations)

This paper contains 8 sections, 29 theorems, 113 equations.

Table of Contents

  1. Introduction
  2. Basic facts about conjugations and spectral measures
  3. Decompositions of conjugations
  4. A matrix description of $\mathscr{C}_s(U)$
  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

Key Result

Lemma 2.3

Suppose $\mathcal{H}$ and $\mathcal{K}$ are Hilbert spaces and $V: \mathcal{H} \to \mathcal{K}$ is a unitary operator. If $C$ is a conjugation on $\mathcal{H}$ then $VCV^{*}$ is a conjugation on $\mathcal{K}$.

Theorems & Definitions (61)

  • Example 2.2
  • Lemma 2.3
  • Example 2.4
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • Proposition 2.7
  • Proposition 2.11
  • proof
  • Remark 2.14
  • ...and 51 more