On the Intertwining of $\partial{\mathcal{D}}$-Isometries

Abstract.

Let ${\mathcal{D}}$ be a strictly pseudoconvex bounded domain in ${\mathbb{C}}^m$ with C ² boundary $\partial{\mathcal{D}}$. If a subnormal m-tuple T of Hilbert space operators has the spectral measure of its minimal normal extension N supported on $\partial{\mathcal{D}}$, then T is referred to as a $\partial{\mathcal{D}}$-isometry. Using some non-trivial approximation theorems in the theory of several complex variables, we establish a commutant lifting theorem for those $\partial{\mathcal{D}}$-isometries whose (joint) Taylor spectra are contained in a special superdomain Ω of ${\mathcal{D}}$. Further, we provide a function-theoretic characterization of those subnormal tuples whose Taylor spectra are contained in Ω and that are quasisimilar to a certain (fixed) $\partial{\mathcal{D}}$-isometry T (of which the multiplication tuple on the Hardy space of the unit ball in ${\mathbb{C}}^m$ is a rather special example).