An Invariant Subspace Theorem and Invariant Subspaces of Analytic Reproducing Kernel Hilbert Spaces - II

详细信息    查看全文

• 作者：Jaydeb Sarkar
• 关键词：Tuples of operators ; Joint invariant subspaces ; Drury–Arveson space ; Weighted Bergman spaces ; Hardy space ; Reproducing kernel Hilbert space ; Multiplier space
• 刊名：Complex Analysis and Operator Theory
• 出版年：2016
• 出版时间：April 2016
• 年：2016
• 卷：10
• 期：4
• 页码：769-782
• 全文大小：457 KB
• 参考文献：1.Aleman, A., Richter, S., Sundberg, C.: Beurlings theorem for the Bergman space. Acta. Math. 177, 275–310 (1996)MathSciNet CrossRef MATH
  2.Arazy, J., Englis, M.: Analytic models for commuting operator tuples on bounded symmetric domains. Trans. Amer. Math. Soc. 355(2), 837–864 (2003)MathSciNet CrossRef MATH
  3.Aronszajn, N.: Theory of reproducing kernels. Trans. Amer. Math. Soc. 68, 337–404 (1950)MathSciNet CrossRef MATH
  4.Arveson, W.: Subalgebras of \(C^*\) -algebras III: multivariable operator theory. Acta. Math. 181, 159–228 (1998)MathSciNet CrossRef MATH
  5.Beurling, A.: On two problems concerning linear transformations in Hilbert space. Acta. Math. 81, 239–255 (1949)MathSciNet CrossRef MATH
  6.Ball, J., Bolotnikov, V., Fang, Q.: Multivariable backward-shift-invariant subspaces and observability operators. Multidimens. Syst. Signal Process 18, 191–248 (2007)MathSciNet CrossRef MATH
  7.Ball, J., Bolotnikov, V.: A Beurling type theorem in weighted Bergman spaces. C. R. Math. Acad. Sci. Paris 351(11–12), 433–436 (2013)MathSciNet CrossRef MATH
  8.Ball, J., Bolotnikov, V.: Weighted Bergman spaces: shift-invariant subspaces and input/state/output linear systems. Integral Equ. Oper. Theory 76, 301–356 (2013)MathSciNet CrossRef MATH
  9.Douglas, R., Misra, G., Sarkar, J.: Contractive Hilbert modules and their dilations. Isr. J. Math. 187, 141–165 (2012)MathSciNet CrossRef MATH
  10.Douglas, R., Paulsen, V.: Hilbert Modules over Function Algebras. In: Research Notes in Mathematics Series, 47. Longman, Harlow (1989)
  11.Greene, D., Richter, S., Sundberg, C.: The structure of inner multipliers on spaces with complete Nevanlinna-Pick kernels. J. Funct. Anal. 194(2), 311–331 (2002)MathSciNet CrossRef MATH
  12.Guo, K., Hu, J., Xu, X.: Toeplitz algebras, subnormal tuples and rigidity on reproducing \({\mathbb{C}}[z_1, \ldots, z_d]\) -modules. J. Funct. Anal. 210(1), 214–247 (2004)MathSciNet CrossRef MATH
  13.Halmos, P.: Shifts on Hilbert spaces. J. Reine Angew. Math. 208, 102–112 (1961)MathSciNet MATH
  14.Lax, P.: Translation invariant spaces. Acta. Math. 101, 163–178 (1959)MathSciNet
  15.McCullough, S., Richter, S.: Bergman-type reproducing kernels, contractive divisors, and dilations. J. Funct. Anal. 190(2), 447–480 (2002)MathSciNet CrossRef MATH
  16.McCullough, S., Trent, T.: Invariant subspaces and Nevanlinna-Pick kernels. J. Funct. Anal. 178(1), 226–249 (2000)MathSciNet CrossRef MATH
  17.Muller, V., Vasilescu, F.-H.: Standard models for some commuting multioperators. Proc. Amer. Math. Soc. 117(4), 979–989 (1993)MathSciNet CrossRef MATH
  18.Sz-Nagy, B., Foias, C.: Harmonic Analysis of Operators on Hilbert Space. North Holland Publishing Company, Amsterdam, Budapest (1970)MATH
  19.Olofsson, A.: A characteristic operator function for the class of \(n\) -hypercontractions. J. Funct. Anal. 236, 517–545 (2006)MathSciNet CrossRef MATH
  20.Radjavi, H., Rosenthal, P.: Invariant subspaces. In: Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 77. Springer, New York (1973)
  21.Sarkar, J.: An invariant subspace theorem and invariant subspaces of analytic reproducing kernel Hilbert spaces - I. J. Oper. Theory 73, 433–441 (2015)MathSciNet CrossRef MATH
  22.Sarkar, J.: An Introduction to Hilbert module approach to multivariable operator theory. In: Alpay D (ed) Operator Theory, pp 969–1033, Springer (2015)
  23.Zhu, K., Zhao, R.: Theory of Bergman spaces on the unit ball. Memoires de la SMF 115, (2008)
  
• 作者单位：Jaydeb Sarkar (1)
  
  1. Indian Statistical Institute, Statistics and Mathematics Unit, 8th Mile, Mysore Road, Bangalore, 560059, India
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  Operator Theory
  Analysis
  
• 出版者：Birkh盲user Basel
• ISSN：1661-8262

文摘

This paper is a follow-up contribution to our work (Sarkar in J Oper Theory, 73:433–441, 2015) where we discussed some invariant subspace results for contractions on Hilbert spaces. Here we extend the results of (Sarkar in J Oper Theory, 73:433–441, 2015) to the context of n-tuples of bounded linear operators on Hilbert spaces. Let \(T = (T_1, \ldots , T_n)\) be a pure commuting co-spherically contractive n-tuple of operators on a Hilbert space \({\mathcal {H}}\) and \({\mathcal {S}}\) be a non-trivial closed subspace of \({\mathcal {H}}\). One of our main results states that: \({\mathcal {S}}\) is a joint T-invariant subspace if and only if there exists a partially isometric operator \(\Pi \in {\mathcal {B}}(H^2_n({\mathcal {E}}), {\mathcal {H}})\) such that \({\mathcal {S}}= \Pi H^2_n({\mathcal {E}})\), where \(H^2_n\) is the Drury–Arveson space and \({\mathcal {E}}\) is a coefficient Hilbert space and \(T_i \Pi = \Pi M_{z_i}\), \(i = 1, \ldots , n\). In particular, it follows that a shift invariant subspace of a “nice” reproducing kernel Hilbert space over the unit ball in \({{\mathbb {C}}}^n\) is the range of a “multiplier” with closed range. Our work addresses the case of joint shift invariant subspaces of the Hardy space and the weighted Bergman spaces over the unit ball in \({{\mathbb {C}}}^n\).