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

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 $\mathrm{S}\mathrm{U}(2)$ and tori are worked out in detail.