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The polar decomposition of the product of three operators

Dingyi Du, Qingxiang Xu, Shuo Zhao

Abstract

In the setting of adjointable operators on Hilbert $C^*$-modules, this paper deals with the polar decomposition of the product of three operators. The relationship between the polar decompositions associated with three operators is clarified. Based on this relationship, a formula for the polar decomposition of a multiplicative perturbation of an operator is provided. In addition, some characterizations of the polar decomposition associated with three operators are provided.

The polar decomposition of the product of three operators

Abstract

In the setting of adjointable operators on Hilbert -modules, this paper deals with the polar decomposition of the product of three operators. The relationship between the polar decompositions associated with three operators is clarified. Based on this relationship, a formula for the polar decomposition of a multiplicative perturbation of an operator is provided. In addition, some characterizations of the polar decomposition associated with three operators are provided.
Paper Structure (4 sections, 19 theorems, 102 equations)

This paper contains 4 sections, 19 theorems, 102 equations.

Table of Contents

  1. Preliminaries
  2. The relationship between the polar decompositions
  3. The polar decomposition of a multiplicative perturbation
  4. The polar decomposition of the product of three operators

Key Result

Lemma 1.1

LLX-AIOT Let $A\in\mathcal{L}(H,K)$ and $B,C\in\mathcal{L}(E,H)$ be such that $\overline{\mathcal{R}(B)}=\overline{\mathcal{R}(C)}$. Then $\overline{\mathcal{R}(AB)}=\overline{\mathcal{R}(AC)}$.

Theorems & Definitions (47)

  • Lemma 1.1
  • Lemma 1.2
  • Lemma 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • ...and 37 more