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Complex Analysis and Operator Theory

, Volume 8, Issue 6, pp 1325–1339 | Cite as

Maximal Contractive Tuples

  • B. Krishna Das
  • Jaydeb Sarkar
  • Santanu SarkarEmail author
Article
  • 148 Downloads

Abstract

Maximality of a contractive tuple of operators is considered. A characterization for a contractive tuple to be maximal is obtained. The notion of maximality for a submodule of the Drury–Arveson module on the \(d\)-dimensional unit ball \({\mathbb {B}}_d\) is defined. For \(d=1\), it is shown that every submodule of the Hardy module over the unit disc is maximal. But for \(d\ge 2\) we prove that any homogeneous submodule or submodule generated by polynomials is not maximal. A characterization of maximal submodules is obtained.

Keywords

Contractive tuples Defect operators Defect spaces  Drury–Arveson module Fock space 

Mathematics Subject Classification (1991)

Primary 15A03 47A13 

Notes

Acknowledgments

We thank the referee for some valuable comments. The hospitality of Indian Statistical Institute, Bangalore centre is warmly and gratefully acknowledged by the first author. The third author was supported by UGC Centre for Advanced Study.

References

  1. 1.
    Arveson, W.: Subalgebras of \(C^{*}\)-algebras III: multivariable operator theory. Acta Math. 181, 159–228 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Bercovici, H.: Operator theory and arithmetic in \(H^{\infty }\). Mathematical Surveys and Monographs. No. 26, A.M.S., Providence (1988)Google Scholar
  3. 3.
    Bhattacharyya, T., Das, B.K., Sarkar, S.: The defect sequence for contractive tuples. Linear Algebra Appl. 438(1), 315–330 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Drury, S.: A generalization of von Neumann’s inequality to the complex ball. Proc. Am. Math. Soc. 68(3), 300–304 (1978)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Gau, H.-L., Wu, P.Y.: Defect indices of powers of a contraction. Linear Algebra Appl. 432, 2824–2833 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Greene, D., Richter, S., Sundberg, C.: The structure of inner multiplications on space with complete Nevanlinna Pick kernels. J. Funct. Anal. 194, 311–331 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    McCullough, S., Trent, T.: Invariant subspaces and Nevanlinna–Pick kernels. J. Funct. Anal. 178, 226–249 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Popescu, G.: Characteristic functions for infinite sequences of noncommuting operators. J. Oper. Theory. 22(1), 51–71 (1989)Google Scholar
  9. 9.
    Popescu, G.: Isometric dilations for infinite sequences of noncommuting operators. Trans. Am. Math. Soc. 316(2), 523–536 (1989)CrossRefzbMATHGoogle Scholar
  10. 10.
    Pott, S.: Standard models under polynomial positivity conditions. J. Oper. Theory. 41, 365–389 (1999)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Sz.-Nagy, B., Foias, C.: Harmonic analysis of operators on Hilbert space. North-Holland, Amsterdam (1970)Google Scholar

Copyright information

© Springer Basel 2013

Authors and Affiliations

  • B. Krishna Das
    • 1
  • Jaydeb Sarkar
    • 1
  • Santanu Sarkar
    • 2
    Email author
  1. 1.Indian Statistical InstituteStatistics and Mathematics UnitBangaloreIndia
  2. 2.Department of MathematicsIndian Institute of ScienceBangaloreIndia

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