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

Abstract

Let D be a strictly pseudoconvex bounded domain in C m with C 2 boundary ∂D. If a subnormal m-tuple T of Hilbert space operators has the spectral measure of its minimal normal extension N supported on ∂D, then T is referred to as a ∂D-isometry. Using some non-trivial approximation theorems in the theory of several complex variables, we establish a commutant lifting theorem for those ∂D-isometries whose (joint) Taylor spectra are contained in a special superdomain Ω of D. Further, we provide a functiontheoretic characterization of those subnormal tuples whose Taylor spectra are contained in Ω and that are quasisimilar to a certain (fixed) ∂D-isometry T (of which the multiplication tuple on the Hardy space of the unit ball in C m is a rather special example). Mathematics Subject Classification (2000). Primary 47B20; Secondary 30E10, 32A30.

Some exact sequences for Toeplitz algebras of spherical isometries

“…C * -algebras generated by isometric representations of commuting semigroups have been studied mainly for semigroups of positive elements in ordered abelian groups; see [BC70], [Do72], [Mu87]. For the case of a single finite spherical isometry the existence of a normal extension along with a commutant lifting theorem were proved in [At90]; see also [AL96] for alternate proofs. A completely different proof of the subnormality of a spherical isometry appears in [Ar98].…”

“…, T z n } on the Hardy space H 2 (S 2n−1 ) of the unit sphere in C n form a spherical isometry which is called the Szegő n-tuple on S 2n−1 . Spherical isometries, mostly with a finite number of components, have been recently studied in a number of papers; see [At90], [At98], [AL96], [Did05], [Es99], [Es01], [Es06], [EsP01]. In [At90] it was proved that every spherical isometry with a finite number of terms is subnormal in the sense that it has a commuting normal extension.…”

“…Spherical isometries, mostly with a finite number of components, have been recently studied in a number of papers; see [At90], [At98], [AL96], [Did05], [Es99], [Es01], [Es06], [EsP01]. In [At90] it was proved that every spherical isometry with a finite number of terms is subnormal in the sense that it has a commuting normal extension. The main purpose of the present paper is to extend this result to an arbitrary family of commuting spherical isometries with a finite or even an infinite number of components.…”

“…The main purpose of the present paper is to extend this result to an arbitrary family of commuting spherical isometries with a finite or even an infinite number of components. Our approach is completely different from that in [At90], and it is mainly based on the study of the space of Toeplitz-type operators associated to spherical isometries. We show that this space is the range of a completely positive idempotent mapping on B(H) and use the properties of this mapping to construct simultaneous normal extensions via the Stinesprig Dilation Theorem.…”

An Introduction to Hilbert Module Approach to Multivariable Operator Theory

“…We refer the reader to Athavale [At92], [At90] for an analytic approach and Attele and Lubin [AtLu96] for a geometric approach to the (regular unitary) dilation theory. In particular, Athavale proved that a spherical isometry must be subnormal.…”