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On a binary operation for positive operators

Shigeru Furuichi, Hamid Reza Moradi, Cristian Conde, Mohammad Sababheh

Abstract

M. Lin defined a binary operation for two positive semi-definite matrices in studying certain determinantal inequalities that arise from diffusion tensor imaging. This operation enjoys some interesting properties similar to the operator geometric mean. We study this operation further and present numerous properties emphasizing the relationship with the operator geometric mean. In the end, we present an application toward Tsallis relative operator entropy.

On a binary operation for positive operators

Abstract

M. Lin defined a binary operation for two positive semi-definite matrices in studying certain determinantal inequalities that arise from diffusion tensor imaging. This operation enjoys some interesting properties similar to the operator geometric mean. We study this operation further and present numerous properties emphasizing the relationship with the operator geometric mean. In the end, we present an application toward Tsallis relative operator entropy.
Paper Structure (4 sections, 27 theorems, 96 equations)

This paper contains 4 sections, 27 theorems, 96 equations.

Table of Contents

  1. Introduction and preliminaries
  2. Main Result
  3. Weighted versions
  4. An application: Quasi-Tsallis relative operator entropy

Key Result

Lemma 1.1

2 Let $G,H\in \mathcal{B}\left( \mathcal{H} \right)$ be positive operators. Then

Theorems & Definitions (56)

  • Lemma 1.1
  • Lemma 1.2
  • Lemma 1.3
  • Example 2.1
  • Proposition 2.1
  • Lemma 2.1
  • proof
  • Theorem 2.1
  • proof
  • Proposition 2.2
  • ...and 46 more