Tirthankar Bhattacharyya

On CNC commuting contractive tuples

The characteristic function has been an important tool for studying completely non-unitary contractions on Hilbert spaces. In this note, we consider completely non-coisometric contractive tuples of commuting operators on a Hilbert space H. We show that the characteristic function, which is now an operator-valued analytic function on the open Euclidean unit ball in ℂn, is a complete unitary invariant for such a tuple. We prove that the characteristic function satisfies a natural transformation law under biholomorphic mappings of the unit ball. We also characterize all operator-valued analytic functions which arise as characteristic functions of pure commuting contractive tuples.

Coherent states on Hilbert modules

We generalize the concept of coherent states, traditionally defined as special families of vectors on Hilbert spaces, to Hilbert modules. We show that Hilbert modules over C*-algebras are the natural settings for a generalization of coherent states defined on Hilbert spaces. We consider those Hilbert C*-modules which have a natural left action from another C*-algebra, say A. The coherent states are well defined in this case and they behave well with respect to the left action by A. Certain classical objects like the Cuntz algebra are related to specific examples of coherent states. Finally we show that coherent states on modules give rise to a completely positive definite kernel between two C*-algebras, in complete analogy to the Hilbert space situation. Related to this, there is a dilation result for positive operator-valued measures, in the sense of Naimark. A number of examples are worked out to illustrate the theory. Some possible physical applications are also mentioned.

Holomorphic functions on the symmetrized bidisk - realization, interpolation and extension

Abstract There are three new things in this paper about the open symmetrized bidisk G = { ( z 1 + z 2 , z 1 z 2 ) : | z 1 | , | z 2 | 1 } . They are, in the order in which they will be proved, (1) The Realization Theorem: A realization formula is demonstrated for every f in the norm unit ball of H ∞ ( G ) . (2) The Interpolation Theorem: A Nevanlinna–Pick interpolation theorem is proved for data from the symmetrized bidisk and a specific formula is obtained for the interpolating function. (3) The Extension Theorem: Let V be a subset of the symmetrized bidisk G . Consider a function f that is holomorphic in a neighbourhood of V and bounded on V. A necessary and sufficient condition on f is obtained so that f possesses an H ∞ -norm preserving extension to the whole of G .

Some thoughts on Ando's theorem and Parrott's example

In this short note, we present an elementary proof of Andos theorem within a restricted class P of homomorphisms modeled after Parrotts example. We also show by explicit estimation that the cb-norm of the contractive homomorphism ρ of the tri-disc algebra, induced by the commuting triple of Parrott, exceeds 1. Indeed, we construct a polynomial P with matrix coefficients with the property ∥ρ(P)∥>∥P∥∞. In particular, we show that there are contractive homomorphisms of the tri-disc algebra which are not even 2-contractive.

Hilbert W*-modules and coherent states

Hilbert C*-module valued coherent states was introduced earlier by Ali, Bhattacharyya and Shyam Roy. We consider the case when the underlying C*-algebra is a W*-algebra. The construction is similar with a substantial gain. The associated reproducing kernel is now algebra valued, rather than taking values in the space of bounded linear operators between two C*-algebras.

On commuting compact self-adjoint operators on a Pontryagin space

Suppose thatA1,A2, ...,An are compact commuting self-adjoint linear maps on a Pontryagin spaceK of indexk and that their joint root subspaceM0 at the zero eigenvalue in ℂn is a nondegenerate subspace. Then there exist joint invariant subspacesH andF inK such thatK=F⊗H,H is a Hilbert space andF is finite-dimensional space withk≤dimF≤(n+2)k. We also consider the structure of restrictionsAj|F in the casek=1.