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C*-algebraic Schur product theorem, Pólya-Szegő-Rudin question and Novak's conjecture

K. Mahesh Krishna

Abstract

Striking result of Vybíral [\textit{Adv. Math.} 2020] says that Schur product of positive matrices is bounded below by the size of the matrix and the row sums of Schur product. Vybíral used this result to prove the Novak's conjecture. In this paper, we define Schur product of matrices over arbitrary C*-algebras and derive the results of Schur and Vybíral. As an application, we state C*-algebraic version of Novak's conjecture and solve it for commutative unital C*-algebras. We formulate Pólya-Szegő-Rudin question for the C*-algebraic Schur product of positive matrices.

C*-algebraic Schur product theorem, Pólya-Szegő-Rudin question and Novak's conjecture

Abstract

Striking result of Vybíral [\textit{Adv. Math.} 2020] says that Schur product of positive matrices is bounded below by the size of the matrix and the row sums of Schur product. Vybíral used this result to prove the Novak's conjecture. In this paper, we define Schur product of matrices over arbitrary C*-algebras and derive the results of Schur and Vybíral. As an application, we state C*-algebraic version of Novak's conjecture and solve it for commutative unital C*-algebras. We formulate Pólya-Szegő-Rudin question for the C*-algebraic Schur product of positive matrices.

Paper Structure

This paper contains 5 sections, 18 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. C*-algebraic Schur product, Schur product theorem and Pólya-Szegő-Rudin question
  3. Lower bounds for C*-algebraic Schur product
  4. C*-algebraic Novak's conjecture
  5. Acknowledgements

Key Result

Theorem 1.1

VYBIRAL Let $A \in M_n(\mathbb{K})$ be a positive matrix. Let $M=AA^*$ and $y\in \mathbb{K}^n$ be the vector of row sums of $A$. Then

Theorems & Definitions (33)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Definition 1.6
  • Lemma 1.7
  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • ...and 23 more