Bhattacharyya, T. ; Sarkar, J. (2006) Characteristic function for polynomially contractive commuting tuples Journal of Mathematical Analysis and Applications, 321 (1). pp. 242259. ISSN 0022247X

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Related URL: http://dx.doi.org/10.1016/j.jmaa.2005.07.075
Abstract
In this note, we develop the theory of characteristic function as an invariant for ntuples 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 Pball (Definition 1.1). Using the reproducing kernel Hilbert space H_{p}(C) associated with this Reinhardt domain in C_{n}, 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 θ_{T} 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 H_{p}(C). It is natural to study the boundary behaviour of θ_{T} 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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