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An Introduction to Hilbert Module Approach to Multivariable Operator Theory

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Operator Theory

Abstract

Let {T 1, …, T n } be a set of n commuting bounded linear operators on a Hilbert space \(\mathcal{H}\). Then the n-tuple (T 1, …, T n ) turns \(\mathcal{H}\) into a module over \(\mathbb{C}[z_{1},\ldots,z_{n}]\) in the following sense:

$$\displaystyle{\mathbb{C}[z_{1},\ldots,z_{n}] \times \mathcal{H}\rightarrow \mathcal{H},\quad \quad (p,h)\mapsto p(T_{1},\ldots,T_{n})h,}$$

where \(p \in \mathbb{C}[z_{1},\ldots,z_{n}]\) and \(h \in \mathcal{H}\). The above module is usually called the Hilbert module over \(\mathbb{C}[z_{1},\ldots,z_{n}]\). Hilbert modules over \(\mathbb{C}[z_{1},\ldots,z_{n}]\) (or natural function algebras) were first introduced by R.G. Douglas and C. Foias in 1976. The two main driving forces were the algebraic and complex geometric views to multivariable operator theory.

This article gives an introduction of Hilbert modules over function algebras and surveys some recent developments. Here the theory of Hilbert modules is presented as combination of commutative algebra, complex geometry and the geometry of Hilbert spaces, and its applications to the theory of n-tuples (n ≥ 1) of commuting operators. The topics which are studied include: model theory from Hilbert module point of view, Hilbert modules of holomorphic functions, module tensor products, localizations, dilations, submodules and quotient modules, free resolutions, curvature, and Fredholm Hilbert modules. More developments in the study of Hilbert module approach to operator theory can be found in a companion paper, “Applications of Hilbert Module Approach to Multivariable Operator Theory.”

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References

  1. Abrahamse, M., Douglas, R.: A class of subnormal operators related to multiply connected domains. Adv. Math. 19, 106–148 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  2. Agler, J.: Rational dilation on an annulus. Ann. Math. 121(2), 537–563; 8(160), 182, 147–163 (1985)

    Google Scholar 

  3. Agler, J., McCarthy, J.: Pick Interpolation and Hilbert Function Spaces. Graduate Studies in Mathematics, vol. 44. American Mathematical Society, Providence (2002)

    Google Scholar 

  4. Agler, J., Young, N.: A model theory for \(\Gamma \)-contractions. J. Oper. Theory 49, 45–60 (2003)

    MathSciNet  MATH  Google Scholar 

  5. Agler, J., Harland, J., Raphael, B.: Classical function theory, operator dilation theory, and machine computation on multiply-connected domains. Mem. Am. Math. Soc. 191, 892 (2008)

    MathSciNet  Google Scholar 

  6. Aleksandrov, A.: The existence of inner functions in a ball. Mat. Sb. 118, 147–163 (1982)

    MathSciNet  Google Scholar 

  7. Aleman, A., Richter, S., Sundberg, C.: Beurling’s theorem for the Bergman space. Acta Math. 177, 275–310 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  8. Alpay, D.: A remark on the Cowen Douglas classes \(B_{n}(\Omega )\). Arch. Math. 51, 539–546 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  9. Anatolii, A., Kaliuzhnyi-Verbovetskyi, D., Vinnikov, V., Woerdeman, H.: Classes of tuples of commuting contractions satisfying the multivariable von Neumann inequality. J. Funct. Anal. 256, 3035–3054 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ando, T.: On a pair of commuting contractions. Acta Sci. Math. (Szeged) 24, 88–90 (1963)

    MathSciNet  MATH  Google Scholar 

  11. Apostol, C., Martin, M.: A C ∗-algebra approach to the Cowen–Douglas theory. In: Topics in Modern Operator Theory (Timisoara/Herculane, 1980). Operator Theory: Advances and Applications, vol. 2, pp. 45–51. Birkhsuser, Basel/Boston (1981)

    Google Scholar 

  12. Apostol, C., Bercovici, H., Foias, C., Pearcy, C.: Invariant subspaces, dilation theory, and the structure of the predual of a dual algebra, I. J. Funct. Anal. 63, 369–404 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  13. Arazy, J., Englis, M.: Analytic models for commuting operator tuples on bounded symmetric domains. Trans. Am. Math. Soc. 355, 837–864 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  14. Aronszajn, N.: Theory of reproducing kernels. Trans. Am. Math. Soc. 68, 337–404 (1950)

    Article  MathSciNet  MATH  Google Scholar 

  15. Arveson, W.: Subalgebras of C ∗-algebras III: multivariable operator theory. Acta Math. 181, 159–228 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  16. Arveson, W.: The curvature invariant of a Hilbert module over \(\mathbb{C}[z_{1},\ldots,z_{n}]\). J. Reine Angew. Math. 522, 173–236 (2000)

    MathSciNet  MATH  Google Scholar 

  17. Athavale, A.: On the intertwining of joint isometries. J. Oper. Theory 23, 339–350 (1990)

    MathSciNet  MATH  Google Scholar 

  18. Athavale, A.: Model theory on the unit ball in \(\mathbb{C}^{m}\). J. Oper. Theory 27, 347–358 (1992)

    MathSciNet  MATH  Google Scholar 

  19. Attele, K., Lubin, A.: Dilations and commutant lifting for jointly isometric operators—a geometric approach. J. Funct. Anal. 140, 300–311 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  20. Bagchi, B., Misra, G.: The homogeneous shifts. J. Funct. Anal. 204, 293–319 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  21. Ball, J.: Rota’s theorem for general functional Hilbert spaces. Proc. Am. Math. Soc. 64, 55–61 (1977)

    MATH  Google Scholar 

  22. Ball, J.: Factorization and model theory for contraction operators with unitary part. Mem. Am. Math. Soc. 13, 198 (1978)

    Google Scholar 

  23. Ball, J.: A lifting theorem for operator models of finite rank on multiply-connected domains. J. Oper. Theory 1, 3–25 (1979)

    MATH  Google Scholar 

  24. Ball, J., Bolotnikov, V.: Weighted Bergman spaces: shift-invariant subspaces and input/state/output linear systems. Int. Equ. Oper. Theory 76, 301–356 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  25. Ball, J., Bolotnikov, V.: A Beurling type theorem in weighted Bergman spaces. C. R. Math. Acad. Sci. Paris 351, 433–436 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  26. Ball, J., Helton, J.W.: A Beurling-Lax theorem for the Lie group U(m,n) which contains most classical interpolation theory. J. Oper. Theory 9, 107–142 (1983)

    MathSciNet  MATH  Google Scholar 

  27. Ball, J., Kriete, T.: Operator-valued Nevanlinna–Pick kernels and the functional models for contraction operators. Int. Equ. Oper. Theory 10, 17–61 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  28. Ball, J., Vinnikov, V.: Lax-Phillips scattering and conservative linear systems: a Cuntz-algebra multidimensional setting. Mem. Am. Math. Soc. 178, 837 (2005)

    MathSciNet  Google Scholar 

  29. Ball, J., Trent, T., Vinnikov, V.: Interpolation and commutant lifting for multipliers on reproducing kernel Hilbert spaces. In: Operator Theory and Analysis (Amsterdam, 1997). Operator Theory: Advances and Applications, vol. 122, pp. 89–138. Birkhäuser, Basel (2001)

    Google Scholar 

  30. Barbian, C.: Beurling-type representation of invariant subspaces in reproducing kernel Hilbert spaces. Int. Equ. Oper. Theory 61, 299–323 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  31. Barbian, C.: A characterization of multiplication operators on reproducing kernel Hilbert spaces. J. Oper. Theory 65, 235–240 (2011)

    MathSciNet  MATH  Google Scholar 

  32. Beauzamy, B.: Introduction to Operator Theory and Invariant Subspaces. North-Holland Mathematical Library, vol. 42. North-Holland, Amsterdam (1988)

    Google Scholar 

  33. Benhida, C., Timotin, D.: Characteristic functions for multicontractions and automorphisms of the unit ball. Int. Equ. Oper. Theory 57, 153–166 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  34. Benhida, C., Timotin, D.: Some automorphism invariance properties for multicontractions. Indiana Univ. Math. J. 56, 481–499 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  35. Beurling, A.: On two problems concerning linear transformations in Hilbert space. Acta Math. 81, 239–255 (1949)

    Article  MATH  Google Scholar 

  36. Bhattacharyya, T., Eschmeier, J., Sarkar, J.: Characteristic function of a pure commuting contractive tuple. Int. Equ. Oper. Theory 53, 23–32 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  37. Biswas, S., Keshari, D., Misra, G.: Infinitely divisible metrics and curvature inequalities for operators in the Cowen–Douglas class. J. Lond. Math. Soc. 88, 941–956 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  38. Bolotnikov, V.: Interpolation for multipliers on reproducing kernel Hilbert spaces. Proc. Am. Math. Soc. 131, 1373–1383 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  39. Burbea, J., Masani, P.: Banach and Hilbert Spaces of Vector-Valued Functions. Their General Theory and Applications to Holomorphy. Research Notes in Mathematics, vol. 90. Pitman Advanced Publishing Program, Boston/London/Melbourne (1984)

    Google Scholar 

  40. Carleson, L.: Interpolations by bounded analytic functions and the corona problem. Ann. Math. 76(2), 547–559 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  41. Carlson, J., Clark, D.: Cohomology and extensions of Hilbert modules. J. Funct. Anal. 128, 278–306 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  42. Carlson, J., Clark, D.: Projectivity and extensions of Hilbert modules over \(A(\mathbb{D}^{n})\). Michigan Math. J. 44, 365–373 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  43. Chalendar, I., Partington, J.: Modern Approaches to the Invariant-Subspace Problem. Cambridge Tracts in Mathematics, vol. 188. Cambridge University Press, Cambridge (2011)

    Google Scholar 

  44. Chattopadhyay, A., Das, B.K., Sarkar, J., Sarkar, S.: Wandering subspaces of the Bergman space and the Dirichlet space over polydisc. Int. Equ. Oper. Theory 79, 567–577 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  45. Chen, L.: On unitary equivalence of quasi-free Hilbert modules. Studia Math. 195, 87–98 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  46. Chen, X., Douglas, R.G.: Localization of Hilbert modules. Mich. Math. J. 39, 443–454 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  47. Chen, X., Guo, K.: Analytic Hilbert Modules. Chapman & Hall/CRC Research Notes in Mathematics, vol. 433. Chapman & Hall/CRC, Boca Raton (2003)

    Google Scholar 

  48. Chen, L., Douglas, R., Guo, K.: On the double commutant of Cowen–Douglas operators. J. Funct. Anal. 260, 1925–1943 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  49. Cowen, M., Douglas, R.: Complex geometry and operator theory. Acta Math. 141, 187–261 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  50. Cowen, M., Douglas, R.: On Operators Possessing an Open Set of Eigen-Values. Memorial Conf. for Fejer-Riesz, Colloq. Math. Soc. J. Bolyai, vol. 35, pp. 323–341, North-Holland, Amsterdam (1983)

    Google Scholar 

  51. Curto, R.: Fredholm invertible n-tuples of operators. The deformation problem. Trans. Am. Math. Soc. 266, 129–159 (1981)

    MathSciNet  MATH  Google Scholar 

  52. Curto, R.: Applications of several complex variables to multiparameter spectral theory. In: Conway, J.B., Morrel, B.B. (eds.) Surveys of Some Recent Results in Operator Theory, vol. II, pp. 25–90. Longman, London (1988)

    Google Scholar 

  53. Curto, R., Herrero, D.: On closures of joint similarity orbits. Int. Equ. Oper. Theory 8, 489–556 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  54. Curto, R., Salinas, N.: Generalized Bergman kernels and the Cowen–Douglas theory. Am. J. Math. 106, 447–488 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  55. Curto, R., Vasilescu, F.-H.: Standard operator models in the polydisc. Indiana Univ. Math. J. 42, 791–810 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  56. Curto, R., Vasilescu, F.-H.: Standard operator models in the polydisc. II. Indiana Univ. Math. J. 44, 727–746 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  57. Davidson, K., Douglas, R.: The generalized Berezin transform and commutator ideals. Pac. J. Math. 222, 29–56 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  58. Didas, M., Eschmeier, J.: Unitary extensions of Hilbert \(A(\mathbb{D})\)-modules split. J. Funct. Anal. 238, 565–577 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  59. Douglas, R.: Hilbert modules over function algebras. In: Advances in Invariant Subspaces and Other Results of Operator Theory (Timisoara and Herculane, 1984). Operator Theory: Advances and Applications, vol. 17, pp. 125–139. Birkhauser, Basel (1986)

    Google Scholar 

  60. Douglas, R.: On Silov resolution of Hilbert modules. Special classes of linear operators and other topics (Bucharest, 1986). Operator Theory: Advances and Applications, vol. 28, pp. 51–60. Birkhauser, Basel (1988)

    Google Scholar 

  61. Douglas, R.: Operator theory and complex geometry. Extracta Math. 24, 135–165 (2009)

    MathSciNet  MATH  Google Scholar 

  62. Douglas, R.: Variations on a theme of Beurling. N. Y. J. Math. 17A, 1–10 (2011)

    MATH  Google Scholar 

  63. Douglas, R.: Connections of the corona problem with operator theory and complex geometry. In: Proceedings of the Fields Institute. arXiv:1212.0455 (to appear)

    Google Scholar 

  64. Douglas, R., Eschmeier, J.: Spectral inclusion theorems. Mathematical methods in systems. In: Optimization, and Control. Operator Theory: Advances and Applications, vol. 222, pp. 113–128. Birkhauser/Springer Basel AG, Basel (2012)

    Google Scholar 

  65. Douglas, R., Foias, C.: A homological view in dilation theory. Preprint series in mathematics, No. 15. Institutul De Matematica, Bucuresti (1976)

    Google Scholar 

  66. Douglas, R., Misra, G.: Quasi-free resolutions of Hilbert modules. Int. Equ. Oper. Theory 47, 435–456 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  67. Douglas, R., Misra, G.: On quasi-free Hilbert modules. N. Y. J. Math. 11, 547–561 (2005)

    MathSciNet  MATH  Google Scholar 

  68. Douglas, R., Paulsen, V.: Hilbert Modules over Function Algebras. Research Notes in Mathematics Series, vol. 47. Longman, Harlow (1989)

    Google Scholar 

  69. Douglas, R., Sarkar, J.: On unitarily equivalent submodules. Indiana Univ. Math. J. 57, 2729–2743 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  70. Douglas, R., Sarkar, J.: Some remarks on the Toeplitz Corona problem. In: Hilbert Spaces of Analytic Functions, CRM Proc. Lecture Notes, vol. 51, pp. 81–89. American Mathematical Society, Providence (2010)

    Google Scholar 

  71. Douglas, R., Sarkar, J.: A note on semi-Fredholm Hilbert modules. Oper. Theory Adv. Appl. 202, 143–150 (2010)

    MathSciNet  Google Scholar 

  72. Douglas, R., Sarkar, J.: Essentially reductive weighted shift Hilbert modules. J. Oper. Theory 65, 379–401 (2011)

    MathSciNet  MATH  Google Scholar 

  73. Douglas, R., Yan, K.: A multi-variable Berger–Shaw theorem. J. Oper. Theory 27, 205–217 (1992)

    MathSciNet  MATH  Google Scholar 

  74. Douglas, R., Yan, K.: Hilbert–Samuel polynomials for Hilbert modules. Indiana Univ. Math. J. 42, 811–820 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  75. Douglas, R., Paulsen, V., Yan, K.: Operator theory and algebraic geometry. Bull. Am. Math. Soc. (N.S.) 20, 67–71 (1989)

    Google Scholar 

  76. Douglas, R., Paulsen, V., Sah, C.-H., Yan, K.: Algebraic reduction and rigidity for Hilbert modules. Am. J. Math. 117, 75–92 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  77. Douglas, R., Misra, G., Varughese, C.: On quotient modules—the case of arbitrary multiplicity. J. Funct. Anal. 174, 364–398 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  78. Douglas, R., Sun, S., Zheng, D.: Multiplication operators on the Bergman space via analytic continuation. Adv. Math. 226, 541–583 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  79. Douglas, R., Kim, Y., Kwon, H., Sarkar, J.: Curvature invariant and generalized canonical operator models: I. Oper. Theory Adv. Appl. 221, 293–304 (2012)

    MathSciNet  Google Scholar 

  80. Douglas, R., Misra, G., Sarkar, J.: Contractive Hilbert modules and their dilations. Israel J. Math. 187, 141–165 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  81. Douglas, R., Putinar, M., Wang, K.: Reducing subspaces for analytic multipliers of the Bergman space. J. Funct. Anal. 263, 1744–1765 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  82. Douglas, R., Kim, Y., Kwon, H., Sarkar, J.: Curvature invariant and generalized canonical operator models: II. J. Funct. Anal. 266, 2486–2502 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  83. Dritschel, M., McCullough, S.: The failure of rational dilation on a triply connected domain. J. Am. Math. Soc. 18, 873–918 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  84. Drury, S.: A generalization of von Neumann’s inequality to the complex ball. Proc. Am. Math. Soc. 68, 300–304 (1978)

    MathSciNet  MATH  Google Scholar 

  85. Eisenbud, D.: Commutative Algebra with a View Toward Algebraic Geometry. Springer, New York (1995)

    MATH  Google Scholar 

  86. Eschmeier, J.: On the Hilbert–Samuel multiplicity of Fredholm tuples. Indiana Univ. Math. J. 56, 1463–1477 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  87. Eschmeier, J.: Samuel multiplicity and Fredholm theory. Math. Ann. 339, 21–35 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  88. Eschmeier, J.: Samuel multiplicity for several commuting operators. J. Oper. Theory 60, 399–414 (2008)

    MathSciNet  MATH  Google Scholar 

  89. Eschmeier, J.: Fredholm spectrum and growth of cohomology groups. Studia Math. 186, 237–249 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  90. Eschmeier, J., Putinar, M.: Spectral Decompositions and Analytic Sheaves. London Mathematical Society Monographs. New Series, vol. 10. Oxford Science Publications/The Clarendon Press/Oxford University Press, New York (1996)

    Google Scholar 

  91. Eschmeier, J., Putinar, M.: Spherical contractions and interpolation problems on the unit ball. J. Reine Angew. Math. 542, 219–236 (2002)

    MathSciNet  MATH  Google Scholar 

  92. Eschmeier, J., Schmitt, J.: Cowen–Douglas operators and dominating sets. J. Operator Theory 72, 277–290 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  93. Fang, X.: Hilbert polynomials and Arveson’s curvature invariant. J. Funct. Anal. 198, 445–464 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  94. Fang, X.: The Fredholm index of a pair of commuting operators. Geom. Funct. Anal. 16, 367–402 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  95. Fang, X.: Estimates for Hilbertian Koszul homology. J. Funct. Anal. 255, 1–12 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  96. Fang, X.: The Fredholm index of a pair of commuting operators, II. J. Funct. Anal. 256, 1669–1692 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  97. Fefferman, C., Stein, E.: H pspaces of several variables. Acta Math. 129, 137–193 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  98. Foias, C., Frazho, A.: The Commutant Lifting Approach to Interpolation Problems. Operator Theory: Advances and Applications, vol. 44. Birkhauser, Basel (1990)

    Google Scholar 

  99. Fuhrmann, P.: A Polynomial Approach to Linear Algebra, 2nd edn. Universitext. Springer, New York (2012)

    Book  MATH  Google Scholar 

  100. Greene, D., Richter, S., Sundberg, C.: The structure of inner multipliers on spaces with complete Nevanlinna–Pick kernels. J. Funct. Anal. 194, 311–331 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  101. Griffiths, P., Harris, J.: Principles of Algebraic Geometry. Wiley Classics Library. Wiley, New York (1994)

    Book  MATH  Google Scholar 

  102. Guo, K.: Normal Hilbert modules over the ball algebra A(B). Studia Math. 135, 1–12 (1999)

    MathSciNet  MATH  Google Scholar 

  103. Guo, K., Huang, H.: Multiplication operators defined by covering maps on the Bergman space: the connection between operator theory and von Neumann algebras. J. Funct. Anal. 260, 1219–1255 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  104. Halmos, P.: Shifts on Hilbert spaces. J. Reine Angew. Math. 208, 102–112 (1961)

    MathSciNet  MATH  Google Scholar 

  105. Halmos, P.: A Hilbert Space Problem Book, 2nd edn. Graduate Texts in Mathematics, vol. 19. Springer, New York/Berlin (1982)

    Google Scholar 

  106. Hedenmalm, H., Perdomo, Y.: Mean value surfaces with prescribed curvature form. J. Math. Pures Appl. 83, 1075–1107 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  107. Helson, H.: Lectures on Invariant Subspaces. Academic, New York/London (1964)

    MATH  Google Scholar 

  108. Hormander, L.: Generators for some rings of analytic functions. Bull. Am. Math. Soc. 73, 943–949 (1967)

    Article  MathSciNet  Google Scholar 

  109. Izuchi, K.J., Izuchi, K.H., Izuchi, Y.: Wandering subspaces and the Beurling type Theorem I. Arch. Math. (Basel) 95, 439–446 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  110. Jewell, N., Lubin, A.: Commuting weighted shifts and analytic function theory in several variables. J. Oper. Theory 1, 207–223 (1979)

    MathSciNet  MATH  Google Scholar 

  111. Jiang, C., Ji, K.: Similarity classification of holomorphic curves. Adv. Math. 215, 446–468 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  112. Jiang, C., Wang, Z.: Structure of Hilbert Space Operators. World Scientific, Hackensack (2006)

    Book  MATH  Google Scholar 

  113. Jiang, C., Guo, X., Ji, K.: K-group and similarity classification of operators. J. Funct. Anal. 225, 167–192 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  114. Jury, M.: Invariant subspaces for a class of complete Pick kernels. Proc. Am. Math. Soc. 133, 3589–3596 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  115. Koranyi, A., Misra, G.: A classification of homogeneous operators in the Cowen–Douglas class. Int. Equ. Oper. Theory 63, 595–599 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  116. Koranyi, A., Misra, G.: A classification of homogeneous operators in the Cowen–Douglas class. Adv. Math. 226, 5338–5360 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  117. Lax, P.: Translation invariant spaces. Acta Math. 101, 163–178 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  118. Lin, Q.: Operator theoretical realization of some geometric notions. Trans. Am. Math. Soc. 305, 353–367 (1988)

    Article  MATH  Google Scholar 

  119. Lubin, A.: Models for commuting contractions. Michigan Math. J. 23, 161–165 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  120. Lubin, A.: Weighted shifts and products of subnormal operators. Indiana Univ. Math. J. 26, 839–845 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  121. Mandrekar, V.: The validity of Beurling theorems in polydiscs. Proc. Am. Math. Soc. 103, 145–148 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  122. Martin, M.: Hermitian geometry and involutive algebras. Math. Z. 188, 359–382 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  123. McCarthy, J.: Boundary values and Cowen–Douglas curvature. J. Funct. Anal. 137, 1–18 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  124. McCullough, S., Richter, S.: Bergman-type reproducing kernels, contractive divisors, and dilations. J. Funct. Anal. 190, 447–480 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  125. McCullough, S., Trent, T.: Invariant subspaces and Nevanlinna–Pick kernels. J. Funct. Anal. 178, 226–249 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  126. Misra, G.: Curvature inequalities and extremal properties of bundle shifts. J. Oper. Theory 11, 305–317 (1984)

    MATH  Google Scholar 

  127. Misra, G., Sastry, N.: Bounded modules, extremal problems, and a curvature inequality. J. Funct. Anal. 88, 118–134 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  128. Muller, V.: Spectral Theory of Linear Operators and Spectral Systems in Banach Algebras. Operator Theory: Advances and Applications, vol. 139. Birkhauser, Basel (2007)

    Google Scholar 

  129. Muller, V., Vasilescu, F.-H.: Standard models for some commuting multioperators. Proc. Am. Math. Soc. 117, 979–989 (1993)

    Article  MathSciNet  Google Scholar 

  130. Nagy, B.S., Foias, C.: Harmonic Analysis of Operators on Hilbert Space. North Holland, Amsterdam (1970)

    MATH  Google Scholar 

  131. Nikolski, N.: Operators, Functions, and Systems: An Easy Reading. Mathematical Surveys and Monographs, vols. 1, 2, pp. 92–93. American Mathematical Society, Providence (2002)

    Google Scholar 

  132. Olofsson, A.: A characteristic operator function for the class of n-hypercontractions. J. Funct. Anal. 236, 517–545 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  133. Parrott, S.: Unitary dilations for commuting contractions. Pac. J. Math. 34, 481–490 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  134. Popescu, G.: Characteristic functions for infinite sequences of noncommuting operators. J. Oper. Theory 22, 51–71 (1989)

    MATH  Google Scholar 

  135. Popescu, G.: Poisson transforms on some C ∗-algebras generated by isometries. J. Funct. Anal. 161, 27–61 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  136. Popescu, G.: Operator theory on noncommutative varieties. Indiana Univ. Math. J. 55, 389–442 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  137. Popescu, G.: Operator theory on noncommutative domains. Mem. Am. Math. Soc. 205, 964 (2010)

    Google Scholar 

  138. Popescu, G.: Joint similarity to operators in noncommutative varieties. Proc. Lond. Math. Soc. 103(3), 331–370 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  139. Putinar, M.: On the rigidity of Bergman submodules. Am. J. Math. 116, 1421–1432 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  140. Radjavi, H., Rosenthal, P.: Invariant Subspaces. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 77. Springer, New York/Heidelberg (1973)

    Google Scholar 

  141. Richter, S.: Unitary equivalence of invariant subspaces of Bergman and Dirichlet spaces. Pac. J. Math. 133, 151–156 (1988)

    Article  MATH  Google Scholar 

  142. Richter, S., Sundberg, C.: Joint extensions in families of contractive commuting operator tuples. J. Funct. Anal. 258, 3319–3346 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  143. Rosenblum, M., Rovnyak, J.: Hardy Classes and Operator Theory. Dover, Mineola (1997). Corrected reprint of the 1985 original

    Google Scholar 

  144. Rota, G.-C.: On models for linear operators. Commun. Pure Appl. Math. 13, 469–472 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  145. Rudin, W.: Function Theory in Polydiscs. Benjamin, New York (1969)

    MATH  Google Scholar 

  146. Rudin, W.: Function Theory in the Unit Ball of \(\mathbb{C}^{n}\). Springer, New York (1980)

    Book  Google Scholar 

  147. Sarkar, J.: Submodules of the Hardy module over polydisc. Israel J. Math. arXiv:1304.1564 (to appear)

    Google Scholar 

  148. Sarkar, J.: An invariant subspace theorem and invariant subspaces of analytic reproducing kernel Hilbert spaces: I. J. Oper. Theory. arXiv:1309.2384 (to appear)

    Google Scholar 

  149. Sarkar, J.: An invariant subspace theorem and invariant subspaces of analytic reproducing kernel Hilbert spaces: II. Preprint. arXiv:1310.1014 (2013)

    Google Scholar 

  150. Sarkar, J.: Applications of Hilbert module approach to multivariable operator theory. In: Alpay, D. (ed.) Operator Theory, chapter 39, pp. 1035–1092, Springer, Basel (2015). doi: 10.1007/978-3-0348-0692-3_69

    Chapter  Google Scholar 

  151. Sarkar, J.: Operator theory on symmetrized bidisc. Indiana Uni. Math. J. http://www.iumj.indiana.edu/IUMJ/Preprints/5541.pdf (to appear)

  152. Sarkar, J.: Wold decomposition for doubly commuting isometries. Linear Algebra Appl. 445, 289–301 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  153. Sarkar, J., Sasane, A., Wick, B.: Doubly commuting submodules of the Hardy module over polydiscs. Studia Math. 217, 179–192 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  154. Shimorin, S.: Wold-type decompositions and wandering subspaces for operators close to isometries. J. Reine Angew. Math. 531, 147–189 (2001)

    MathSciNet  MATH  Google Scholar 

  155. Stegenga, D.: Multipliers of the dirichlet space. Ill. J. Math. 24, 113–139 (1980)

    MathSciNet  MATH  Google Scholar 

  156. Taylor, J.: The analytic-functional calculus for several commuting operators. Acta Math. 125, 1–38 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  157. Taylor, J.: A joint spectrum for several commuting operators. J. Funct. Anal. 6, 172–191 (1970)

    Article  MATH  Google Scholar 

  158. Uchiyama, M.: Curvatures and similarity of operators with holomorphic eigenvectors. Trans. Am. Math. Soc. 319, 405–415 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  159. Varopoulos, N.: On an inequality of von Neumann and an application of the metric theory of tensor products to operators theory. J. Funct. Anal. 16, 83–100 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  160. Vasilescu, F.-H.: Analytic functional calculus and spectral decompositions. Translated from the Romanian. Mathematics and its Applications (East European Series), vol. 1. D. Reidel Publishing, Dordrecht (1982); Editura Academiei Republicii Socialiste România, Bucharest 

    Google Scholar 

  161. Vasilescu, F.-H.: An operator-valued Poisson kernel. J. Funct. Anal. 110, 47–72 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  162. Venugopalkrishna, U.: Fredholm operators associated with strongly pseudoconvex domains in \(\mathbb{C}^{n}\). J. Funct. Anal. 9, 349–373 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  163. von Neumann, J.: Eine Spektraltheorie fur allgemeine Operatoren eines unitdren Raumes. Math. Nachr. 4, 258–281 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  164. Wells, R.O.: Differential Analysis on Complex Manifolds. Graduate Texts in Mathematics. Springer, New York/Berlin (1980)

    Book  MATH  Google Scholar 

  165. Wold, H.: A Study in the Analysis of Stationary Time Series. Almquist and Wiksell, Uppsala (1938)

    MATH  Google Scholar 

  166. Wolff, R.: Spectra of analytic Toeplitz tuples on Hardy spaces. Bull. Lond. Math. Soc. 29, 65–72 (1997)

    Article  Google Scholar 

  167. Yan, K.: Equivalence of homogeneous principal Hardy submodules. Indiana Univ. Math. J. 41, 875–884 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  168. Zhu, K.: Mobius invariant Hilbert spaces of holomorphic functions in the unit ball of \(\mathbb{C}^{n}\). Trans. Am. Math. Soc. 323, 823–842 (1991)

    MATH  Google Scholar 

  169. Zhu, K.: Operators in Cowen–Douglas classes. Ill. J. Math. 44, 767–783 (2000)

    MATH  Google Scholar 

  170. Zhao, R., Zhu, K.: Theory of Bergman spaces in the unit ball of \(\mathbb{C}^{n}\). Mém. Soc. Math. France (N.S.) No. 115 (2008)

    Google Scholar 

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  1. Statistics and Mathematics Unit, Indian Statistical Institute, Bangalore, India

    Jaydeb Sarkar

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  1. Earl Katz Chair in Algebraic System Theory, Department of Mathematics, Ben-Gurion University of the Negev, Be’er Sheva, Israel

    Daniel Alpay

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Sarkar, J. (2015). An Introduction to Hilbert Module Approach to Multivariable Operator Theory. In: Alpay, D. (eds) Operator Theory. Springer, Basel. https://doi.org/10.1007/978-3-0348-0667-1_59

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