Separately radial and radial Toeplitz operators on the projective space and representation theory

Abstract

We consider separately radial (with corresponding group Tⁿ) and radial (with corresponding group U(n)) symbols on the projective space Pⁿ(C), as well as the associated Toeplitz operators on the weighted Bergman spaces. It is known that the C*-algebras generated by each family of such Toeplitz operators are commutative (see R. Quiroga-Barranco and A. Sanchez-Nungaray (2011)). We present a new representation theoretic proof of such commutativity. Our method is easier and more enlightening as it shows that the commutativity of the C*-algebras is a consequence of the existence of multiplicity-free representations. Furthermore, our method shows how to extend the current formulas for the spectra of the corresponding Toeplitz operators to any closed group lying between Tⁿ and U(n).