Separately Radial and Radial Toeplitz Operators on the Projective Space and Representation Theory

Abstract

We consider separately radial (with corresponding group ${\mathbb{T}}^n$) and radial (with corresponding group $\mathrm{U}(n))$ symbols on the projective space ${\mathbb{P}}^n(\mathbb{C})$, as well as the associated Toeplitz operators on the weighted Bergman spaces. It is known that the $C^{\ast }$-algebras generated by each family of such Toeplitz operators are commutative. 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 $C^{\ast }$-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 ${\mathbb{T}}^n$ and $\mathrm{U}(n)$.

Corrected version

References in corpus (1)

• Separately Radial and Radial Toeplitz Operators on the Unit Ball and Representation Theory