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A Haagerup inequality through the use of orthogonal polynomials

Félix Parraud

Abstract

In this paper we prove that the Haagerup inequality for non-homogeneous polynomials in free semicircular variables of degree $n$ is optimal with a constant of order $n^{3/2}$. We also show an operator valued Haagerup inequality which improves on existing results. Our main tool to do so are free Chebyshev polynomials also known as $0$-Hermite polynomials.

A Haagerup inequality through the use of orthogonal polynomials

Abstract

In this paper we prove that the Haagerup inequality for non-homogeneous polynomials in free semicircular variables of degree is optimal with a constant of order . We also show an operator valued Haagerup inequality which improves on existing results. Our main tool to do so are free Chebyshev polynomials also known as -Hermite polynomials.
Paper Structure (5 sections, 16 theorems, 72 equations)

This paper contains 5 sections, 16 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Definition and properties of the free Chebyshev polynomials
  4. The Haagerup inequality
  5. The operator valued version

Key Result

Theorem 1.1

There exists a family of orthogonal projections $(P_n)_{n\geq 0}$ such that for any given $z\in L^2(x)$, if $\sum_{n\geq 0} (n+1) \left\Vert P_nz\right\Vert_2$ is finite, then $z\in\mathcal{C}^*(x)$, and In particular, $P_0+\dots+P_n$ is the orthogonal projection on the set of polynomials of degree at most $n$. Hence for any such polynomials $P$, Besides one can find polynomials $Q_n$ of degree

Theorems & Definitions (36)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Definition 3.1
  • Definition 3.2
  • ...and 26 more