Szegő limit theorems for Toeplitz operators on compact homogeneous spaces

Abstract:

Let $f$ be a real valued integrable function on a compact homogeneous space $M = K\backslash G$ and ${M_f}$ the operator of pointwise multiplication by $f$. The authors consider families of Toeplitz operators ${T_{f,P}} = P{M_f}P$ as $P$ ranges over a net of orthogonal projections from ${L^2}(M)$ to finite dimensional $G$-invariant subspaces. Necessary and sufficient conditions are given on the net in order that the distribution of eigenvalues of these Toeplitz operators is asymptotic to the distribution of values of $f$ in the sense of Szegö’s classical theorem for the circle. Explicit sequences satisfying these conditions are constructed for all compact Lie groups and for all Riemannian symmetric compact spaces.