Journal of the Mathematical Society of Japan

Common reducing subspaces of several weighted shifts with operator weights

Caixing GU

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Abstract

We characterize common reducing subspaces of several weighted shifts with operator weights. As applications, we study the common reducing subspaces of the multiplication operators by powers of coordinate functions on Hilbert spaces of holomorphic functions in several variables. The identification of reducing subspaces also leads to structure theorems for the commutants of von Neumann algebras generated by these multiplication operators. This general approach applies to weighted Hardy spaces, weighted Bergman spaces, Drury–Arveson spaces and Dirichlet spaces of the unit ball or polydisk uniformly.

Article information

Source
J. Math. Soc. Japan, Volume 70, Number 3 (2018), 1185-1225.

Dates
Received: 16 March 2016
Revised: 31 January 2017
First available in Project Euclid: 25 June 2018

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1529892025

Digital Object Identifier
doi:10.2969/jmsj/74677467

Mathematical Reviews number (MathSciNet)
MR3830805

Zentralblatt MATH identifier
06966980

Subjects
Primary: 47B37: Operators on special spaces (weighted shifts, operators on sequence spaces, etc.) 47B35: Toeplitz operators, Hankel operators, Wiener-Hopf operators [See also 45P05, 47G10 for other integral operators; see also 32A25, 32M15] 47A15: Invariant subspaces [See also 47A46]
Secondary: 46E22: Hilbert spaces with reproducing kernels (= [proper] functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) [See also 47B32] 32A35: Hp-spaces, Nevanlinna spaces [See also 32M15, 42B30, 43A85, 46J15] 32A36: Bergman spaces

Keywords
weighted shifts with operator weights reducing subspaces analytic Toeplitz operators weighted Hardy space on unit ball weighted Bergman spaces Dirichlet space on polydisk

Citation

GU, Caixing. Common reducing subspaces of several weighted shifts with operator weights. J. Math. Soc. Japan 70 (2018), no. 3, 1185--1225. doi:10.2969/jmsj/74677467. https://projecteuclid.org/euclid.jmsj/1529892025


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References

  • M. B. Abrahamse and J. A. Ball, Analytic Toeplitz operators with automorphic symbol II, Proc. Amer. Math. Soc., 59 (1976), 323–328.
  • H. Bercovici, C. Foias and C. Pearcy, Dual algebra with applications to invariant subspaces and dilation theory, CBMS Regional Conference Series in Mathematics, 56, 1985.
  • A. Beurling, On two problems concerning linear transformation in Hilbert space, Acta Math., 81 (1949), 239–255.
  • J. B. Conway, A Course in Operator Theory, Graduate Studies in Math., 21, Amer. Math. Soc., Providence, Rhode Island, 1999.
  • C. Gu, Reducing subspaces of weighted shifts with operator weights, Bull. Korean Math. Soc., 53 (2016), 1471–1481.
  • K. Guo and H. Huang, Multiplication operators on the Bergman space, Lecture Notes in Math., 2145, Springer, 2015.
  • H. Hedenmalm, B. Korenblum and K. Zhu, Theory of Bergman Spaces, GTM, 199, Springer-Verlag, 2000.
  • H. Hedenmalm, S. Richter and K. Seip, Interpolating sequences and invariant subspaces of given index in the Bergman spaces, J. Reine Angew. Math., 477 (1996), 13–30.
  • P. Halmos, Shifts on Hilbert spaces, J. Reine Angew. Math., 208 (1961), 102–112.
  • K. J. Izuchi, T. Nakazi and M. Seto, Backward shift invariant subspaces in the bidisc, II, J. Operator Theory, 51 (2004), 361–376.
  • K. J. Izuchi, K. H. Izuchi and Y. Izuchi, Wandering subspaces and the Beurling type theorem, III, J. Math. Soc. Japan, 64 (2012), 627–658.
  • N. P. Jewell and A. R. Lubin, Commuting weighted shifts and analytic function theory in several variables, J. Operator Theory, 1 (1979), 207–223.
  • H. T. Kaptanoğlu, Möbius-invariant Hilbert spaces in polydisk, Pacific J. Math., 163 (1994), 337–360.
  • S. Kuwahara, Reducing subspaces of weighted Hardy spaces on polydisks, Nihonkai Math. J., 25 (2014), 77–83.
  • Y. Lu and X. Zhou, Invariant subspaces and reducing subspaces of weighted Bergman space over the bidisk, J. Math. Soc. Japan, 62 (2010), 745–765.
  • E. Nordgren, Reducing subspaces of analytic Toeplitz operators, Duke Math. J., 34 (1967), 175–181.
  • H. Radjavi and P. Rosenthal, Simultaneous Triangularization, Universitext, Springer-Verlag, 2000.
  • Y. Shi and Y. Lu, Reducing subspaces for Toeplitz operators on the polydisk, Bull. Korean Math. Soc., 50 (2013), 687–696.
  • Y. Shi and N. Zhou, Reducing subspaces of some multiplication operators on the Bergman space over polydisk, Abstract and Applied Analysis, 2015, Art. ID 209307, 12 pp.
  • A. L. Shields, Weighted shift operators and analytic function theory, Math. Surv., 13, Amer. Math. Soc., Providence, 1974, pp. 49–128.
  • M. Stessin and K. Zhu, Reducing subspaces of weighted shift operators, Proc. Amer. Math. Soc., 130 (2002), 2631–2639.
  • Y. Qin and R. Yang, A characterization of submodules via the Beurling–Lax–Halmos theorem, Proc. Amer. Math. Soc., 142 (2014), 3505–3510.
  • K. Zhu, Reducing subspaces for a class of multiplication operators, J. London Math. Soc. (2), 62 (2000), 553–568.