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Oppenheim-Schur's inequality and RKHS

Akira Yamada

Abstract

In 2012, we obtained Oppenheim's inequality for positive semidefinite matrices and its equality condition by the reproducing kernel method. In this paper, as a continuation, we give a reproducing kernel proof of the block matrix version of the Oppenheim-Schur's inequality and its equality condition in the positive definite case.

Oppenheim-Schur's inequality and RKHS

Abstract

In 2012, we obtained Oppenheim's inequality for positive semidefinite matrices and its equality condition by the reproducing kernel method. In this paper, as a continuation, we give a reproducing kernel proof of the block matrix version of the Oppenheim-Schur's inequality and its equality condition in the positive definite case.
Paper Structure (5 sections, 18 theorems, 81 equations)

This paper contains 5 sections, 18 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Interpolation problem and RKHS
  3. Hadamard product of vector-valued RKHSs and inner product interpolation
  4. Main results
  5. Application to determinant inequalities

Key Result

Theorem 1.1

The following inequalities hold:

Theorems & Definitions (38)

  • Theorem 1.1: e.g. Oppenheim30
  • Definition 2.1
  • Definition 2.2: Image of a RKHS
  • Theorem 2.1: Schwartz64*Proposition 21
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.1
  • Lemma 2.3
  • ...and 28 more