Multivariable isometries related to certain convex domains

TL;DR

This paper introduces a new class of convex domains in complex space that differ from traditional classes and explores their connection to subnormal operator tuples, advancing understanding in operator theory and complex geometry.

Contribution

The paper defines the class mega^{(n)} of convex domains in n, distinct from known classes, and applies operator theory techniques to analyze associated subnormal operator tuples.

Findings

Identification of the class mega^{(n)} of convex domains

Application of inner function and -Neumann theories to these domains

New insights into subnormal operator tuples related to these domains

Abstract

There exist several interesting results in the literature on subnormal operator tuples having their spectral properties tied to the geometry of strictly pseudoconvex domains or to that of bounded symmetric domains in $\C^n$. We introduce a class $\Omega^{(n)}$ of convex domains in $\C^n$ which, for $n \geq 2$, is distinct from the class of strictly pseudoconvex domains and the class of bounded symmetric domains and which lends itself for the application of the theories related to the abstract inner function problem and the $\bar \partial$-Neumann problem, allowing us to make a number of interesting observations about certain subnormal operator tuples associated with the members of the class $\Omega^{(n)}$.