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Remarks on positive definite functions on a group

Swapan Jana, Sourav Pal, Nitin Tomar

Abstract

We study the operator-valued positive definite functions on a group using positive block matrices. We give an alternative proof to Brehmer positivity for doubly commuting contractions. We classify all commuting unitary representations over a finite group. We show by examples that the power of a positive-definite function may not be positive definite and also the power of a unitary representation may not be a unitary representation. We also characterize all unitary representations whose powers are also unitary representations.

Remarks on positive definite functions on a group

Abstract

We study the operator-valued positive definite functions on a group using positive block matrices. We give an alternative proof to Brehmer positivity for doubly commuting contractions. We classify all commuting unitary representations over a finite group. We show by examples that the power of a positive-definite function may not be positive definite and also the power of a unitary representation may not be a unitary representation. We also characterize all unitary representations whose powers are also unitary representations.
Paper Structure (9 sections, 40 theorems, 109 equations)

This paper contains 9 sections, 40 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Some elementary results
  3. Positive definite functions on small groups and block matrices
  4. Positive definite functions on group of order 2
  5. Positive definite functions on a group of order 3
  6. Positive definite functions on groups of order 4
  7. Positive Definite Functions on $\mathbb{Z}$ and direct sum of its copies
  8. Structure Theorem for Unitary representations on a finite group
  9. Powers of positive definite functions and unitary representations

Key Result

Theorem 1.1

If $T$ is a contraction acting on a Hilbert space $\mathcal{H}$, then there exist a Hilbert space $\mathcal{K} \supseteq \mathcal{H}$ and a unitary $U$ on $\mathcal{K}$ such that for all $k \in \mathbb{N}\cup \{0\}$. Moreover, $\mathcal{K}$ can be chosen to be minimal in the sense that $\mathcal{K}$ is the smallest closed reducing subspace for $U$ that contains $\mathcal{H}$.

Theorems & Definitions (79)

  • Theorem 1.1: Sz.-Nagy, NagyFoias
  • Definition 1.2
  • Theorem 1.3: Naimark, NeumarkI
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.3: Paulsen, Theorem 4.8
  • Proposition 2.4
  • proof
  • ...and 69 more