Characteristic function for polynomially contractive commuting tuples

Bhattacharyya, T. ; Sarkar, J. (2006) Characteristic function for polynomially contractive commuting tuples Journal of Mathematical Analysis and Applications, 321 (1). pp. 242-259. ISSN 0022-247X

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In this note, we develop the theory of characteristic function as an invariant for n-tuples of operators. The operator tuple has a certain contractivity condition put on it. This condition and the class of domains in Cn that we consider are intimately related. A typical example of such a domain is the open Euclidean unit ball. Given a polynomial P in C[z1,z2,…,zn] whose constant term is zero, all the coefficients are nonnegative and the coefficients of the linear terms are nonzero, one can naturally associate a Reinhardt domain with it, which we call the P-ball (Definition 1.1). Using the reproducing kernel Hilbert space Hp(C) associated with this Reinhardt domain in Cn, S. Pott constructed the dilation for a polynomially contractive commuting tuple (Definition 1.2) [S. Pott, Standard models under polynomial positivity conditions, J. Operator Theory 41 (1999) 365–389. MR 2000j:47019]. Given any polynomially contractive commuting tuple T we define its characteristic function θT which is a multiplier. We construct a functional model using the characteristic function. Exploiting the model, we show that the characteristic function is a complete unitary invariant when the tuple is pure. The characteristic function gives newer and simpler proofs of a couple of known results: one of them is the invariance of the curvature invariant and the other is a Beurling theorem for the canonical operator tuple on Hp(C). It is natural to study the boundary behaviour of θT in the case when the domain is the Euclidean unit ball. We do that and here essential differences with the single operator situation are brought out.

Item Type:Article
Source:Copyright of this article belongs to Elsevier Science.
Keywords:Multivariable Operator Theory; Dilation Theory; Characteristic Function; Model Theory
ID Code:99684
Deposited On:27 Nov 2016 12:53
Last Modified:27 Nov 2016 12:53

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