On CNC commuting contractive tuples

  • T. BhattacharyyaEmail author
  • J. Eschmeier
  • J. Sarkar


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.


Characteristic function invariant subspaces biholomorphic automorphisms functional model coincidence 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Ambrozie C and Eschmeier J, A commutant lifting theorem on analytic polyhedra (2005) (Banach Center Publ.) vol. 67, pp. 83–108Google Scholar
  2. [2]
    Arveson W B, Subalgebras of C*-algebrasIII, Multivariable operator theory,Acta Math. 181(2) (1998) 159–228, MR 2000e: 47013zbMATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    Arveson W, The curvature invariant of a Hilbert module over C[Z1,..., Zd],J. Reine Angew. Math. 522 (2000) 173–236, MR 2003a: 47013zbMATHMathSciNetGoogle Scholar
  4. [4]
    Athavale A, On the intertwining of joint isometries,J. Op. Theory 23(2) (1990) 339–350, MR 91i: 47029zbMATHMathSciNetGoogle Scholar
  5. [5]
    Ball J A and Trent T T, Unitary colligations, reproducing kernel Hilbert spaces, and Nevanlinna-Pick interpolation in several variables,J. Funct. Anal. 157(1) (1998) 1–61, MR 2000b: 47028zbMATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    Ball J A, Trent T T and Vinnikov V, Interpolation and commutant lifting for multipliers on reproducing kernrel Hilbert spaces, in: Operator Theory and Analyisis, OT 122 (2001) (Birkhauser: Bassel) pp. 89–138Google Scholar
  7. [7]
    Benhida C and Timotin D, Characteristic functions for multicontractions and automorphisms of the unit ball, http:/ Scholar
  8. [8]
    Bhattacharyya T, Eschmeier J and Sarkar J, Characteristic function of a pure commuting contractive tuple,Integral Equations Op. Theory 53 (2005) 23–32zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    Bhattacharyya T and Sarkar J, Characteristic function for polynomially contractive commuting tuples,J. Math. Anal. Appl. 321 (2006) 241–259CrossRefMathSciNetGoogle Scholar
  10. [10]
    Bunce J W, Models for n-tuples of noncommuting operators,J. Funct. Anal. 57 (1984) 21–30, MR 85k: 47019zbMATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    Davis C, Some dilation and representation theorems, Proceedings of the Second International Symposium in West Africa on Functional Analysis and its Applications (Kumasi, 1979), Forum forFunct.Anal. Appl. (Kumasi, Ghana, 1979)pp. 159–182, MR 84e: 47012Google Scholar
  12. [12]
    Drury S W, A generalization of von Neumann’s inequality to the complex ball,Proc. Am. Math. Soc. 68(3) (1978) 300–304, MR 80c: 47010zbMATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    Eschmeier J and Putinar M, Spherical contractions and interpolation problems on the unit ball,J. Reine Angew. Math. 542 (2002) 219–236, MR 2002k: 47019zbMATHMathSciNetGoogle Scholar
  14. [14]
    Frazho A E, Models for noncommuting operators,J. Funct. Anal. 48 (1982) 1–11, MR 84h: 47010zbMATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    Greene D C V, Richter S and Sundberg C, The structure of inner multipliers on spaces with complete Nevanlinna Pick kernels,J. Funct. Anal. 194 (2002) 311–331, MR 2003h: 46038zbMATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    McCullough S and Trent T T, Invariant subspaces and Nevanlinna-Pick kernels,J. Funct. Anal. 178 (2000) 226–249, MR 2002b: 47006zbMATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    Muller V and Vasilescu F-H, Standard models for some commuting multioperators,Proc. Am. Math. Soc. 117 (1993) 979–989, MR 93e: 47016CrossRefMathSciNetGoogle Scholar
  18. [18]
    Popescu G, Characteristic functions for infinite sequences of noncommuting operators,J. Op. Theory 22 (1989) 51–71, MR 91m: 47012zbMATHGoogle Scholar
  19. [19]
    Popescu G, Poisson transforms on some C*-algebras generated by isometries,J. Funct. Anal. 161 (1999) 27–61, MR 2000m: 46117zbMATHCrossRefMathSciNetGoogle Scholar
  20. [20]
    Popescu G, Operator theory on non commutative varietiesI,Indiana Univ. Math. J. 55(2) (2006) 389–422zbMATHCrossRefMathSciNetGoogle Scholar
  21. [21]
    Popescu G, Operator theory on non commutative varietiesII,Proc. Am. Math. Soc., to appearGoogle Scholar
  22. [22]
    Rudin W, Function theory in the unit ball of ℂn (1980) (New York: Springer) MR 82i: 32002Google Scholar
  23. [23]
    Sz-Nagy B and Foias C, Harmonic analysis of operators on Hilbert space (1970) (North-Holland) MR 43: 947Google Scholar

Copyright information

© Indian Academy of Sciences 2006

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of ScienceBangaloreIndia
  2. 2.Fachbereich MathematikUniversität des SaarlandesSaarbrückenGermany

Personalised recommendations