14 research outputs found
The Restriction Principle and Commuting Families of Toeplitz Operators on the Unit Ball
On the unit ball B^n we consider the weighted Bergman spaces H_\lambda and
their Toeplitz operators with bounded symbols. It is known from our previous
work that if a closed subgroup H of \widetilde{\SU(n,1)} has a
multiplicity-free restriction for the holomorphic discrete series of
\widetilde{\SU(n,1)}, then the family of Toeplitz operators with H-invariant
symbols pairwise commute. In this work we consider the case of maximal abelian
subgroups of \widetilde{\SU(n,1)} and provide a detailed proof of the pairwise
commutativity of the corresponding Toeplitz operators. To achieve this we
explicitly develop the restriction principle for each (conjugacy class of)
maximal abelian subgroup and obtain the corresponding Segal-Bargmann transform.
In particular, we obtain a multiplicity one result for the restriction of the
holomorphic discrete series to all maximal abelian subgroups. We also observe
that the Segal-Bargman transform is (up to a unitary transformation) a
convolution operator against a function that we write down explicitly for each
case. This can be used to obtain the explicit simultaneous diagonalization of
Toeplitz operators whose symbols are invariant by one of these maximal abelian
subgroups
Commutative -algebras of Toeplitz operators on complex projective spaces
We prove the existence of commutative -algebras of Toeplitz operators on
every weighted Bergman space over the complex projective space
. The symbols that define our algebras are those that
depend only on the radial part of the homogeneous coordinates. The algebras
presented have an associated pair of Lagrangian foliations with distinguished
geometric properties and are closely related to the geometry of
Toeplitz operators on the domain with -invariant symbols
Let be the irreducible bounded symmetric domain of complex
matrices that satisfy . The biholomorphism group of is realized
by with isotropy at the origin given by
. Denote by the subgroup of
diagonal matrices in . We prove that the set of
-invariant essentially bounded symbols yield
Toeplitz operators that generate commutative -algebras on all weighted
Bergman spaces over . Using tools from representation theory, we also
provide an integral formula for the spectra of these Toeplitz operators