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 KΩ(T ) and MΩ(T ). These quantities are related to the quantities KX(T ) and MX(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 Caratheodory metric and the Mobius 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.