Hostname: page-component-5b777bbd6c-cp4x8 Total loading time: 0 Render date: 2025-06-19T06:19:15.695Z Has data issue: false hasContentIssue false
  • English
  • Français

Beurling type invariant subspaces on Hardy and Bergman spaces of the unit ball or polydisk

Part of: General theory of linear operators Special classes of linear operators Linear function spaces and their duals

Published online by Cambridge University Press:  08 January 2025

Caixing Gu Open the ORCID record for Caixing Gu [Opens in a new window] ,
Shuaibing Luo  and
Pan Ma
Caixing Gu
Affiliation:
Department of Mathematics, California Polytechnic State University, San Luis O-bispo, CA, 93407, USA e-mail: cgu@calpoly.edu
Shuaibing Luo
Affiliation:
School of Mathematics, Hunan University, Changsha, 410082, PR China e-mail: sluo@hnu.edu.cn
Pan Ma*
Affiliation:
School of Mathematics and Statistics, HNP-LAMA, Central South University, Changsha, 410083, PR China
*
e-mail: pan.ma@csu.edu.cn
Get access Rights & Permissions [Opens in a new window]

Abstract

McCullough and Trent generalize Beurling–Lax–Halmos invariant subspace theorem for the shift on Hardy space of the unit disk to the multi-shift on Drury–Arveson space of the unit ball by representing an invariant subspace of the multi-shift as the range of a multiplication operator that is a partial isometry. By using their method, we obtain similar representations for a class of invariant subspaces of the multi-shifts on Hardy and Bergman spaces of the unit ball or polydisk. Our results are surprisingly general and include several important classes of invariant subspaces on the unit ball or polydisk.

Keywords

Invariant subspaceBeurling’s theoremreproducing kernel Hilbert space on unit ballpolydisk or polyballdoubly commuting

MSC classification

Primary: 47A15: Invariant subspaces 47B32: Operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces) 46E22: Hilbert spaces with reproducing kernels (=
Type
Article
Information
Canadian Mathematical Bulletin , Volume 68 , Issue 1 , March 2025 , pp. 232 - 245
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

S.L. was supported by the NNSFC (Grant No. 12271149) and Natural Science Foundation of Hunan Province (Grant No. 2024JJ2008). P.M. was supported by the NNSF of China (Grant Nos. 11801572 and 12171484), the Natural Science Foundation of Hunan Province (Grant No. 2023JJ20056), the Science and Technology Innovation Program of Hunan Province (Grant No. 2023RC3028), and Central South University Innovation-Driven Research Programme (Grant No. 2023CXQD032).

References

Agler, J., Hypercontractions and subnormality . J. Operator Theory 13(1985), no. 2, 203–217.Google Scholar
Agler, J. and McCarthy, J. E., Pick interpolation and Hilbert function spaces, Graduate Studies in Mathematics, 44, American Mathematical Society, Providence, Rhode Island, 2002.CrossRefGoogle Scholar
Agrawal, O., Clark, D. N., and Douglas, R. G., Invariant subspaces in the polydisk . Pacific J. Math. 121(1986), 1–11.CrossRefGoogle Scholar
Aleman, A., Richter, S., and Sundberg, C., Beurling’s theorem for the Bergman space . Acta Math. 177(1996), no. 2, 275–310.CrossRefGoogle Scholar
Arveson, W., The curvature invariant of a Hilbert module over $\mathbb{C}[{z}_1,\dots, {z}_d]$ $.$ J. Reine Angew. Math. 522(2000), 173–236.Google Scholar
Ball, J. and Bolotnikov, V., A Beurling type theorem in weighted Bergman spaces . C. R. Math. Acad. Sci. Paris 351(2013), 433–436.CrossRefGoogle Scholar
Ball, J. and Bolotnikov, V., Weighted Bergman spaces: Shift-invariant subspaces and input/state/output linear systems . Integ. Equ. Oper. Theory 76(2013), 301–356.CrossRefGoogle Scholar
Bhattacharjee, M., Eschmeier, J., Keshari, D. K., and Sarkar, J., Dilations, wandering subspaces, and inner functions . Linear Algebra Appl. 523(2017), 263–280.CrossRefGoogle Scholar
Chattopadhyay, A., Das, B. K., Sarkar, J., and Sarkar, S., Wandering subspaces of the Bergman space and the Dirichlet space over ${D}^n$ . Integ. Equ. Oper. Theory 79(2014), no. 4, 567–577.CrossRefGoogle Scholar
Clouâtre, R., Hartz, M., and Schillo, D., A Beurling–Lax–Halmos theorem for spaces with a complete Nevanlinna–Pick factor . Proc. Amer. Math. Soc. 148(2020), 731–740.CrossRefGoogle Scholar
Fricain, E. and Mashreghi, J., The theory of $H(b)$ spaces. Vols. 1, 2, New Mathematical Monographs, 20, 21, Cambridge University Press, Cambridge, 2016.Google Scholar
Gaşpar, D. and Suciu, N., Wold decompositions for commutative families of isometries . An. Univ. Timisoara Ser. Stiint. Mat. 27(1989), 31–38.Google Scholar
Greene, D., Richter, S., and Sundberg, C., The structure of inner multipliers on spaces with complete Nevanlinna–Pick kernels . J. Funct. Anal. 194(2002), 311–331.CrossRefGoogle Scholar
Gu, C., On $(m,p)$ -expansive and $(m,p)$ -contractive operators on Hilbert and Banach spaces . J. Math. Anal. Appl. 426(2015), 893–916.CrossRefGoogle Scholar
Gu, C. and Chen, Z., A model for $(n,p)$ -hypercontractions on Banach space . Indag. Math. (N.S.). 26(2015), no. 3, 485–494.CrossRefGoogle Scholar
Gu, C. and Park, J., Finite zero-based invariant subspaces of the shift operator on reproducing kernel spaces . Complex Var. Elliptic Equ. 67(2022), no. 10, 2519–2544.CrossRefGoogle Scholar
Hedenmalm, H. and Zhu, K., On the failure of optimal factorization for certain weighted Bergman spaces . Complex Var. Th. Appl. 19(1992), 165–176.Google Scholar
Izuchi, K. H., Cyclicity of reproducing kernels in weighted Hardy spaces over the bidisc . J. Funct. Anal. 272(2017), 546–558.CrossRefGoogle Scholar
Izuchi, K. J., Izuchi, K. H., and Izuchi, Y., Ranks of invariant subspaces of the Hardy space over the bidisk . J. Reine Angew. Math. 659(2011), 101–139.Google Scholar
Izuchi, K. J., Nakazi, T., and Seto, M., Backward shift invariant subspaces in the bidisc II . J. Operator Theory 51(2004), 361–376.Google Scholar
Kumari, R., Sarkar, J., Sarkar, S., and Timotin, D., Factorizations of kernels and reproducing kernel Hilbert spaces . Integ. Equ. Oper. Theory 87(2017), no. 2, 225–244.CrossRefGoogle Scholar
Mandrekar, V., The validity of Beurling theorems in polydisk . Proc. Amer. Math. Soc. 103(1988), 145–148.CrossRefGoogle Scholar
McCullough, S. and Richter, S., Bergman-type reproducing kernels, contractive divisors, and dilations . J. Funct. Anal. 190(2002), 447–480.CrossRefGoogle Scholar
McCullough, S. and Trent, T., Invariant subspaces and Nevanlinna–Pick kernels . J. Funct. Anal. 178(2000), 226–249.CrossRefGoogle Scholar
Muller, V. and Vasilescu, F.-H., Standard models for some commuting multioperators . Proc. Amer. Math. Soc. 117(1993), 979–989.CrossRefGoogle Scholar
Olofsson, A., A characteristic operator function for the class of $n$ -hypercontractions . J. Funct. Anal. 236(2006), 517–545.CrossRefGoogle Scholar
Paulsen, V. I. and Raghupathi, M., An introduction to the theory of reproducing kernel Hilbert spaces, Cambridge Studies in Advanced Mathematics, 152, Cambridge University Press, Cambridge, 2016.CrossRefGoogle Scholar
Radlow, J., Ideals of square summable series in several variables . Proc. Amer. Math. Soc. 38(1973), 293–297.CrossRefGoogle Scholar
Redett, D. and Tung, J., Invariant subspaces in Bergman space over the bidisc . Proc. Amer. Math. Soc. 138(2010), no. 7, 2425–2430.CrossRefGoogle Scholar
Richter, S., Invariant subspaces of the Dirichlet shift . J. Reine Angew. Math. 386(1988), 205–220.Google Scholar
Sarkar, J., Wold decomposition for doubly commuting isometries . Linear Algebra Appl. 445(2014), 289–301.CrossRefGoogle Scholar
Sarkar, J., Submodules of the Hardy module over polydisc . Israel J. Math. 205(2015), no. 1, 317–336.CrossRefGoogle Scholar
Sarkar, J., An invariant subspace theorem and invariant subspaces of analytic reproducing kernel Hilbert spaces-I . J. Operator Theory 73(2015), 433–441.CrossRefGoogle Scholar
Sarkar, J., An invariant subspace theorem and invariant subspaces of analytic reproducing kernel Hilbert spaces-II . Complex Anal. Oper. Theory 10(2016), no. 4, 769–782.CrossRefGoogle Scholar
Sarkar, J., Sasane, A., and Wick, B. D., Doubly commuting submodules of the Hardy module over polydiscs . Stud. Math. 217(2013), 179–192.CrossRefGoogle Scholar
Słociński, M., On the Wold-type decompositions of a pair of commuting isometries . Ann. Polon. Math. 37(1980), 255–262.CrossRefGoogle Scholar
Timotin, D., Regular dilations and models for multicontractions . Indiana Univ. Math. J. 47(1998), 671–684.CrossRefGoogle Scholar
Yang, R., A brief survey of operator theory in ${H}^2({D}^2)$ $.$ In: K. Zhu (ed.), Handbook of analytic operator theory, CRC Press/Chapman Hall Handbooks in Mathematics Series, CRC Press, Boca Raton, FL, 2019, pp. 223–258.CrossRefGoogle Scholar