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$μ$-Hankel Operators on Non-Abelian Compact Lie Groups

Emma Sulaver

TL;DR

The paper develops a non-commutative generalization of $\mu$-Hankel operators on compact Lie groups by leveraging the Peter–Weyl decomposition to define matrix-valued symbols and representation weights. The operator $A_{\mu,a}:H^2(G)\to H^2_-(G)$ is introduced with a block structure $T_{\pi,\rho}=\mu(\pi)a(\pi,\rho)\nu(\rho)$, and sharp boundedness, compactness, Schatten class, and Fredholm criteria are established in terms of symbol decay in the classes $\mathcal{S}^{m,n}(\mu,\nu)$. An inverse problem for symbol recovery from spectral data is developed, proving uniqueness under a band-limited assumption and providing stability via Tikhonov regularization; the framework is concretely illustrated on $\mathrm{SU}(2)$ and torus product groups. The results generalize the classical abelian theory, uncovering new phenomena due to non-abelian representation multiplicities and offering a foundation for further non-compact or time–frequency extensions.

Abstract

We introduce and study a natural non-commutative generalization of \(μ\)-Hankel operators originally defined on Hardy spaces over compact abelian groups. Within the framework of Peter-Weyl theory, we define matrix-valued Hankel operators associated to pairs of irreducible representations and weight functions, then establish sharp boundedness and compactness criteria in terms of symbol decay. We characterize membership in Schatten-von Neumann ideals and compute Fredholm indices in key cases. Finally, we initiate the inverse problem of symbol recovery by spectral data, proving uniqueness and stability under mild assumptions. Several illustrative examples on \(\mathrm{SU}(2)\) and tori are worked out in detail.

$μ$-Hankel Operators on Non-Abelian Compact Lie Groups

TL;DR

The paper develops a non-commutative generalization of -Hankel operators on compact Lie groups by leveraging the Peter–Weyl decomposition to define matrix-valued symbols and representation weights. The operator is introduced with a block structure , and sharp boundedness, compactness, Schatten class, and Fredholm criteria are established in terms of symbol decay in the classes . An inverse problem for symbol recovery from spectral data is developed, proving uniqueness under a band-limited assumption and providing stability via Tikhonov regularization; the framework is concretely illustrated on and torus product groups. The results generalize the classical abelian theory, uncovering new phenomena due to non-abelian representation multiplicities and offering a foundation for further non-compact or time–frequency extensions.

Abstract

We introduce and study a natural non-commutative generalization of -Hankel operators originally defined on Hardy spaces over compact abelian groups. Within the framework of Peter-Weyl theory, we define matrix-valued Hankel operators associated to pairs of irreducible representations and weight functions, then establish sharp boundedness and compactness criteria in terms of symbol decay. We characterize membership in Schatten-von Neumann ideals and compute Fredholm indices in key cases. Finally, we initiate the inverse problem of symbol recovery by spectral data, proving uniqueness and stability under mild assumptions. Several illustrative examples on \(\mathrm{SU}(2)\) and tori are worked out in detail.
Paper Structure (7 sections, 18 theorems, 83 equations)

This paper contains 7 sections, 18 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Definition and Basic Properties of $\mu$-Hankel Operators
  4. Boundedness and Compactness Criteria
  5. Schatten–von Neumann and Fredholm Properties
  6. Inverse Problems: Symbol Recovery
  7. Conclusion and Further Directions

Key Result

Theorem 2.3

As a unitary representation of $G\times G$ by left– and right–translation, and the matrix coefficients $\{\sqrt{d_\pi}\,\pi_{ij}:1\le i,j\le d_\pi,\ \pi\in\widehat{G}\}$ form an orthonormal basis of $L^2(G)$.

Theorems & Definitions (49)

  • Definition 2.1: Compact Lie group
  • Definition 2.2: Unitary dual
  • Theorem 2.3: Peter–Weyl Folland2016
  • Lemma 2.4: Schur orthogonality RuzhanskyTurunen2010
  • Definition 2.5: Group Fourier transform
  • Definition 2.6: Hardy spaces on $G$
  • Definition 2.7: Weight functions and symbols Mirotin2023
  • Definition 2.8: Abelian $\mu$-Hankel operator Mirotin2023
  • Definition 2.9: Matrix-valued symbol and weight
  • Definition 2.10: Symbol class $\mathcal{S}(\mu,\nu)$
  • ...and 39 more