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Complex Analysis and Operator Theory

, Volume 6, Issue 1, pp 91–103 | Cite as

Abstract Characteristic Function

  • Tirthankar BhattacharyyaEmail author
Article
  • 74 Downloads

Abstract

The characteristic function for a contraction is a classical complete unitary invariant devised by Sz.-Nagy and Foias. Just as a contraction is related to the Szego kernel \({k_S(z,w) = (1 - z {\overline {w}})^{-1}}\) for |z|, |w| < 1, by means of (1/k S )(T, T*) ≥ 0, we consider an arbitrary open connected domain Ω in \({{\mathbb {C}}^n}\), a kernel k on Ω so that 1/k is a polynomial and a tuple T = (T 1, T 2, . . . , T n ) of commuting bounded operators on a complex separable Hilbert space \({\mathcal H}\) such that (1/k)(T, T*) ≥ 0. Under some standard assumptions on k, it turns out that whether a characteristic function can be associated with T or not depends not only on T, but also on the kernel k. We give a necessary and sufficient condition. When this condition is satisfied, a functional model can be constructed. Moreover, the characteristic function then is a complete unitary invariant for a suitable class of tuples T.

Keywords

Hilbert Space Characteristic Function Polynomial Ring Functional Model Dense Subspace 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Birkhäuser / Springer Basel AG 2010

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of ScienceBangaloreIndia

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