Multivariable Operator Theory on Noncommutative Domains
by
Gelu Popescu
University of Texas San Antonio

We study noncommutative domains D_f ⊂ B(H)ⁿ generated by positive regular free holomorphic functions f, where B(H) is the algebra of all bounded linear operators on a Hilbert space H. Each such a domain has a universal model (W₁, ..., W_n) of weighted shifts acting on the full Fock space with n generators. The study of D_f is close related to the study of the weighted shifts W₁, ..., W_n, their joint invariant subspaces, and the representations of the algebras they generate: the domain algebra A_n(D_f), the Hardy algebra F_n^∞(D_f), and the C^*-algebra C^*(W₁, ..., W_n). A good part of the talk deals with these issues. We discuss problems related to the dilation theory, model theory, and unitary invariants on noncommutative domains and noncommutative varieties. Commutant lifting results and applications are also considered.

Date received: May 28, 2008