Essential commutants on strongly pseudo-convex domains

Consider a bounded strongly pseudo-convex domain Q with smooth boundary in C-n. Let tau be the Toeplitz algebra on the Bergman space L-a(2)(Omega). That is, tau is the C*-algebra generated by the Toeplitz operators {T-f : f is an element of L-infinity(Omega)}. Extending the work [27,28] in the special case of the unit ball, we show that on any such Omega, tau and {T-f: is an element of VObdd} + K are essential commutants of each other, where K is the collection of compact operators on L-a(2)(Omega). On a general Q considered in this paper, the proofs require many new ideas and techniques. These same techniques also enable us to show that for A is an element of tau, if < A(kz), k(z)> -> 0 as z -> partial derivative Omega, then A is a compact operator. (C) 2020 Elsevier Inc. All rights reserved.

Automated summary

This paper examines a bounded strongly pseudo-convex domain and the Toeplitz algebra on the Bergman space. It shows certain properties on specific Omegas and uses new ideas and techniques to derive general conclusions. The findings highlight the essential commutants of operators in different settings, as well as the compactness of certain operators under specific conditions.