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Eigenvalue accumulation for operator convolutions on locally compact groups

Florian Schroth

Abstract

Within the framework of quantum harmonic analysis, for a locally compact group $G$ with a square-integrable, irreducible unitary representation, we analyze the eigenvalue distributions of convolutions between indicator functions on $G$ and a fixed density operator on the representation space, a concept which generalizes localization operators. In particular, we consider a sequence of such operators and the asymptotic number of eigenvalues that lie within a small distance of $1$. We show that a previously postulated type of asymptotic behavior occurs if and only if the group is unimodular and the sets underlying the indicator functions form a Følner sequence. Applying this, we obtain positive results for nilpotent and homogeneous Lie groups, recovering an established result for the Heisenberg group as a special case.

Eigenvalue accumulation for operator convolutions on locally compact groups

Abstract

Within the framework of quantum harmonic analysis, for a locally compact group with a square-integrable, irreducible unitary representation, we analyze the eigenvalue distributions of convolutions between indicator functions on and a fixed density operator on the representation space, a concept which generalizes localization operators. In particular, we consider a sequence of such operators and the asymptotic number of eigenvalues that lie within a small distance of . We show that a previously postulated type of asymptotic behavior occurs if and only if the group is unimodular and the sets underlying the indicator functions form a Følner sequence. Applying this, we obtain positive results for nilpotent and homogeneous Lie groups, recovering an established result for the Heisenberg group as a special case.
Paper Structure (14 sections, 19 theorems, 92 equations)

This paper contains 14 sections, 19 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Notation
  3. Trace class operators on a Hilbert space
  4. Abstract harmonic analysis on locally compact groups
  5. Nilpotent Lie groups
  6. Homogeneous groups
  7. Følner sequences
  8. Operator convolutions
  9. Operator-operator convolutions
  10. Function-operator convolutions
  11. Main result
  12. Applications to nilpotent Lie groups
  13. Application to word metric balls
  14. Application to dilations

Key Result

Theorem 2.1

An operator $S \in {\mathcal{B}(\mathcal{H})}$ belongs to ${\mathcal{S}^1(\mathcal{H})}$ if and only if there exist orthonormal bases $(\psi_n)_{n \in \mathbb{N}}$ and $(\varphi_n)_{n \in \mathbb{N}}$ and a non-increasing sequence $(s_n)_{n \in \mathbb{N}} \in \ell^1(\mathbb{N})$ of non-negative rea with strong operator convergence. In this case, $(s_n)_{n \in \mathbb{N}}$ is unique, eq:svd conver

Theorems & Definitions (40)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Definition 3.1
  • Proposition 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof : Proof of Proposition \ref{['prop:folnerequiv']}
  • ...and 30 more