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Lebesgue decomposition for positive operators revisited

Yoshiki Aibara, Yoshimichi Ueda

Abstract

We explain how Pusz--Woronowicz's idea of their functional calculus fits the theory of Lebesgue decomposition for positive operators on Hilbert spaces initially developed by Ando. In this way, we reconstruct the essential and fundamental part of the theory.

Lebesgue decomposition for positive operators revisited

Abstract

We explain how Pusz--Woronowicz's idea of their functional calculus fits the theory of Lebesgue decomposition for positive operators on Hilbert spaces initially developed by Ando. In this way, we reconstruct the essential and fundamental part of the theory.
Paper Structure (6 sections, 21 theorems, 66 equations)

This paper contains 6 sections, 21 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Pusz--Woronowicz functional calculus
  3. Lebesgue decomposition of positive operators
  4. Radon--Nikodym derivatives
  5. A detailed account of Remark \ref{['R2.5']}
  6. Continuity of PW-functional calculus

Key Result

Lemma 2.1

$R_{A,B} + S_{A,B} = 1_{\mathcal{H}_{A,B}}$. In particular, $(R_{A,B},S_{A,B})$ is a commuting pair of positive bounded operators on $\mathcal{H}_{A,B}$.

Theorems & Definitions (46)

  • Lemma 2.1: PW75; see also HU21
  • Lemma 2.2
  • proof
  • Theorem 2.4
  • proof
  • Remark 2.5
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • proof
  • ...and 36 more