Complex Analysis and Operator Theory

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

Abstract Characteristic Function

  • Tirthankar BhattacharyyaEmail author


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.


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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  1. 1.
    Agler J.: The Arveson extension theorem and coanalytic models. Integral Equ. Oper. Theory 5(5), 608–631 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Agler, J., McCarthy, J.E.: Pick interpolation and Hilbert function spaces. Graduate Studies in Mathematics, vol. 44. American Mathematical Society, Providence (2002)Google Scholar
  3. 3.
    Ambrozie C.-G., Englis M., Müller V.: Operator tuples and analytic models over general domains in \({\mathbb C^n}\). J. Oper. Theory 47(2), 287–302 (2002)zbMATHGoogle Scholar
  4. 4.
    Arveson W.: Subalgebras of C*-algebras. III. Multivariable operator theory. Acta Math. 181(2), 159–228 (1998)MathSciNetGoogle Scholar
  5. 5.
    Arveson W.: The curvature invariant of a Hilbert module over C[z 1, . . . , z d]. J. Reine Angew. Math. 522, 173–236 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Athavale A.: Model theory on the unit ball in C m. J. Oper. Theory 27(2), 347–358 (1992)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Bhattacharyya T., Eschmeier J., Sarkar J.: Characteristic function of a pure commuting contractive tuple. Integral Equ. Oper. Theory 53(1), 23–32 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Bhattacharyya T., Sarkar J.: Characteristic function for polynomially contractive commuting tuples. J. Math. Anal. Appl. 321(1), 242–259 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Bhattacharyya T., Eschmeier J., Sarkar J.: On CNC commuting contractive tuples. Proc. Indian Acad. Sci. Math. Sci. 116(3), 299–316 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Drury S.W.: A generalization of von Neumann’s inequality to the complex ball. Proc. Am. Math. Soc. 68, 300–304 (1978)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Gheondea A., Popescu G.: Bounded characteristic functions and models for noncontractive sequences of operators. Integral Equ. Oper. Theory 45(1), 15–38 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Müller V., Vasilescu F.-H.: Standard models for some commuting multioperators. Proc. Am. Math. Soc. 117(4), 979–989 (1993)zbMATHCrossRefGoogle Scholar
  13. 13.
    Popescu G.: Characteristic functions for infinite sequences of noncommuting operators. J. Oper. Theory 22(1), 51–71 (1989)zbMATHGoogle Scholar
  14. 14.
    Popescu G.: Poisson transforms on some C*-algebras generated by isometries. J. Funct. Anal. 161(1), 27–61 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Popescu G.: Characteristic functions and joint invariant subspaces. J. Funct. Anal. 237(1), 277–320 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Pott S.: Standard models under polynomial positivity conditions. J. Oper. Theory 41(2), 365–389 (1999)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Rudin W.: Functional Analysis. McGraw-Hill, New York (1991)zbMATHGoogle Scholar
  18. 18.
    Sz.-Nagy, B., Foias, C.: Harmonic analysis of operators on Hilbert space. Translated from the French and revised. North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York; Akadmiai Kiad, Budapest (1970)Google Scholar

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© Birkhäuser / Springer Basel AG 2010

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of ScienceBangaloreIndia

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