On the Multiplication Tuples Related to Certain Reproducing Kernel Hilbert Spaces

We consider a class of subnormal operator tuples consisting of multiplications by coordinate functions on a class of reproducing kernel Hilbert spaces associated with certain bounded domains in , with the closure of being the Taylor spectrum of and the topological boundary of being the Taylor essential spectrum of . If T is a subnormal operator tuple quasisimilar to , then we show that the Taylor spectrum of T is provided is polynomially convex and provided is either strictly pseudoconvex with boundary or is starlike, and that the Taylor essential spectrum of T is provided satisfies the Gleason property as well. This generalizes some previous work of the first-named author in the context of the unit ball and the unit polydisk. The relevant theory is then applied to the multiplication tuples on the Hardy and Bergman spaces of complex ellipsoids