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Factorization of anti-linear and $C$-normal operators

Sudip Ranjan Bhuia

Abstract

A conjugation $C$ is an anti-linear isometric involution on a complex Hilbert space $\clh$, and $T\in \clb(\clh)$ is conjugate normal if $T^*T = CTT^*C$ holds for some conjugation (C). In this paper, we provide a factorization and range inclusion theorem for anti-linear operators, and consequently, establish the polar decomposition for anti-linear operators by applying the Douglas theorem on majorization of Hilbert space operators. Moreover, we present a factorization of $C$-normal operators based on the polar decomposition. Lastly, we study the Cartesian decomposition of conjugate normal operators, thereby expanding the results in [18].

Factorization of anti-linear and $C$-normal operators

Abstract

A conjugation is an anti-linear isometric involution on a complex Hilbert space , and is conjugate normal if holds for some conjugation (C). In this paper, we provide a factorization and range inclusion theorem for anti-linear operators, and consequently, establish the polar decomposition for anti-linear operators by applying the Douglas theorem on majorization of Hilbert space operators. Moreover, we present a factorization of -normal operators based on the polar decomposition. Lastly, we study the Cartesian decomposition of conjugate normal operators, thereby expanding the results in [18].
Paper Structure (8 sections, 30 theorems, 51 equations)

This paper contains 8 sections, 30 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Factorization and range inclusion theorem for anti-linear operators
  4. Factorization of conjugate normal operators
  5. Factorization of $C$-normal operators
  6. Cartesian decomposition of $C$-normal operators revisited
  7. Singular value inequalities for compact $C$-normal operators
  8. Some inequalities for self-commutator

Key Result

Theorem 2.1

Douglas Lemma If $A, B \in \mathcal{B}(\mathcal{H})$, then the following statements are equivalent: Moreover, if (i), (ii), and (iii) are valid, then there exists a unique operator $C$ so that

Theorems & Definitions (61)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 2.1: Douglas Theorem
  • Lemma 2.2
  • Theorem 2.3
  • Lemma 2.4
  • Remark 2.5
  • Theorem 2.6
  • Remark 2.7
  • ...and 51 more