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Extremal Schur Multipliers

Erik Christensen

Abstract

The Schur product of two complex m x n matrices is their entry wise product. We show that an extremal element X in the convex set of m x n complex matrices of Schur multiplier norm at most 1 satisfies the inequality rank(X) =< (m +n)^(1/2) . For positive n x n matrices with unit diagonal, we give a characterization of the extremal elements, and show that such a matrix satisfies rank(X) =< n^(1/2).

Extremal Schur Multipliers

Abstract

The Schur product of two complex m x n matrices is their entry wise product. We show that an extremal element X in the convex set of m x n complex matrices of Schur multiplier norm at most 1 satisfies the inequality rank(X) =< (m +n)^(1/2) . For positive n x n matrices with unit diagonal, we give a characterization of the extremal elements, and show that such a matrix satisfies rank(X) =< n^(1/2).

Paper Structure

This paper contains 6 sections, 9 theorems, 12 equations.

Table of Contents

  1. Introduction
  2. Some properties of the Schur product
  3. On a full set of vectors
  4. Extremal positive Schur multipliers of norm 1
  5. Extremal points in the set of Schur multipliers of norm at most 1
  6. An example

Key Result

Theorem 1.1

There exists a positive constant $K^{\mathbb C}_G$ such that for any natural number $n$ and any complex $n \times n$ matrix $X$ of Schur multiplier norm 1, there exists a natural number $k$ and for each $i,$ with $1 \leq i \leq k$ there exist a positive real $t_i$ and vectors $L_i, R_i$ in ${\mathbb

Theorems & Definitions (18)

  • Theorem 1.1
  • Definition 2.1
  • Theorem 2.2
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Theorem 4.1
  • proof
  • ...and 8 more