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

Abstract

Abstract. We consider separately radial (with corresponding group T n ) and radial (with corresponding group U(n)) symbols on the projective space P n (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 [8]). 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 T n and U(n).

Toeplitz Operators with Horizontal Symbols Acting on the Poly-Fock Spaces

“…It has also been adapted to other contexts such as projective spaces, Fock spaces and Cartan domains, among others. We refer to [1,2,5,6,7,8,9,11,12,13,15,16] for examples of research on Toeplitz operators with separately radial, radial symbols, and other types of symbols.…”

“…For the sake of completeness we present, in the next two remarks, a summary of the ideas discussed by Quiroga-Barranco in [14], see also [15,17].…”

“…The proof of Proposition 1 uses the fact that the measure vα is U(n)-invariant and so Sn ⋊ T n -invariant as well, where U (n) is the unitary group (see [4,14,15]).…”

“…The existence of these types of symbols on the unit ball on C n , as well as in other bounded domains, as invariant functions with respect to symmetric subgroups of the biholomorphism group of the corresponding domain was used to generate commutative C * -algebras of Toeplitz operators as we can see in [9,12,15]. These results were based on the relationship between the analytic and the representation theoretic approach to the study of Toeplitz operators whose symbols have symmetries.…”

“…The principal goal in this work is to explore the relationship between the representation theory, as established in [14,15], and the commutativity of the C * -algebra generated for the Toeplitz operators with symbols invariant under the groups Sn ⋊ T n and T n ⋊ An, where T n stands for the n-dimensional torus, Sn stands for the symmetric group of permutations of the set {1, 2, . .…”