Toeplitz Operators on Odd Dimensional Manifolds and Weighted Bergman Kernels with Bounded Toeplitz Products on the Unit Ball.

Abstract

We characterize the finite rank of Bergman and Bargmann Toeplitz operators generated by a measure in many dimensions and show some applications. We showthe finite rank commutator of Toeplitz and Hankel operators with the semicommutators of Toeplitz operators with bounded harmonic symbols .We study the relation of Toeplitz family operators to the higher spectral flow on dimensional manifolds with boundary. We study also the CR invariants and describe the boundary singularity of weighted Bergman kernels. We discuss the distribution functions inequality and investigate the products of Toeplitz operators and Hankel operators on Bergman spaces of the unit ball and on the polydisks.