$\mu$-Hankel Operators on Non-Abelian Compact Lie Groups

TL;DR

This paper generalizes -Hankel operators to non-commutative compact Lie groups, establishing boundedness, compactness, Schatten class membership, and inverse spectral problems within the Peter-Weyl framework.

Contribution

It introduces matrix-valued Hankel operators on non-abelian groups, providing criteria for boundedness, compactness, and spectral inverse problems, extending classical theory to new non-commutative settings.

Findings

Established sharp boundedness and compactness criteria.

Characterized Schatten-von Neumann class membership.

Solved inverse symbol recovery problem.

Abstract

We introduce and study a natural non-commutative generalization of \(\mu\)-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.