Commuting Toeplitz operators on bounded symmetric domains and multiplicity-free restrictions of holomorphic discrete series

Abstract

For any given bounded symmetric domain, we prove the existence of commutative C * -algebras generated by Toeplitz operators acting on any weighted Bergman space. The symbols of the Toeplitz operators that generate such algebras are defined by essentially bounded functions invariant under suitable subgroups of the group of biholomorphisms of the domain. These subgroups include the maximal compact groups of biholomorphisms. We prove the commutativity of the Toeplitz operators by considering the Bergman spaces as the underlying space of the holomorphic discrete series and then applying known multiplicity-free results for restrictions to certain subgroups of the holomorphic discrete series. In the compact case we completely characterize the subgroups that define invariant symbols that yield commuting Toeplitz operators in terms of the multiplicity-free property.1991 Mathematics Subject Classification. Primary: 47B35, 22E46, Secondary: 32A36, 32M15, 22E45.Key words and phrases. Toeplitz operator, Bergman space, holomorphic discrete series, multiplicity-free representation.M. Dawson would like to thank the VIGRE program at LSU, DMS-0739382, for support. He would also like to thank CIMAT for its hospitality and support during a visit in September 2013.

The restriction principle and commuting families of Toeplitz operators on the unit ball

“…Commutative families of Toeplitz operators and representation theory. In this section we briefly review some of the results in [4] which connect commutativity of Toeplitz operators with representation theory, in particular restriction of the discrete series representation π λ to subgroups of G or G.…”

“…For instance, in [4] it was shown that if H ⊂ G is a subgroup such that the restriction π λ | H is multiplicity-free, then the Toeplitz operators over the weighted Bergmann space H 2 λ (B n ) with H-invariant symbols form a commuting family. Many examples of subgroups H ⊂ G that give rise to multiplicity-free restrictions can be found using the method of visible actions developed by Kobayashi (see [10,11,12]; the earlier paper [7] by Faraut and Thomas was also important in the development of this theory).…”

“…For instance, any symmetric subgroup H ⊂ G gives rise to a multiplicity-free restriction ( [12]). Furthermore, it was also shown in [4] that if H ⊂ G is a compact subgroup, then the H-invariant symbols give rise to a commutative family of Toeplitz operators if and only if π λ | H is multiplicity-free. One interesting consequence of these results was that in the case of higher-rank symmetric domains, these families do not correspond to symbols that are invariant under maximal abelian subgroups, as had previously been conjectured, but rather to subgroups of the group of biholomorphisms of the symmetric domain that admit multiplicity-free restrictions of the corresponding scalar-type holomorphic discreteseries representations.…”

Radial Toeplitz operators on the weighted Bergman spaces of Cartan domains

“…Proposition 4.6 (DQO [1]). For a given bounded symbol ψ ∈ L ∞ (D) and a fixed λ > p − 1 the following conditions are equivalent (1) The symbol ψ is radial.…”

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

“…This is particularly so given the abundance of commutative algebras generated by Toeplitz operators on complex spaces. With this respect, the results from [1] have special relevance as they prove the existence of nontrivial commutative C * -algebras generated by Toeplitz operators on every weighted Bergman space over every irreducible bounded symmetric domain. The first fundamental ingredient to prove the latter is one which has become very important for Toeplitz operators in the last years: group theory.…”