Spectral domains in several complex variables

In this paper we study the concepts of spectral domain and complete spectral domain in several complex variables. For a domain Ω in Cⁿ and an n-tuple T of commuting operators on a Hilbert space H such that the Taylor spectrum of T is a subset of Ω, we introduce the quantities K_Ω(T) and M_Ω(T). These quantities are related to the quantities K_X(T) and M_X(T) introduced by Paulsen for a compact subset X. When T is an n-tuple of 2×2 matrices, KΩ(T) and MΩ(T) are expressed in terms of the Carathéodory metric and the Möbius distance. This in turn answers a question by Paulsen for tuples of 2×2 matrices. We also establish von Neumann’s inequality for an n-tuple of upper triangular Toeplitz matrices. We study the regularity of KΩ(T) and MΩ(T) and obtain various comparisons of these two quantities when T is an n-tuple of Jordan blocks.