Multivariable isometries related to certain convex domains

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 ${\text{C}}^n$. We introduce a class ${\mathrm{\Omega }}^{(n)}$ of convex domains in ${\text{C}}^n$ which, for $n\ge 2$, 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 $\bar{\mathrm{\partial }}$-Neumann problem, allowing us to make a number of interesting observations about certain subnormal operator tuples associated with the members of the class ${\mathrm{\Omega }}^{(n)}$.