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An Operator Theoretic Setting of Singly-Generated Invariant Subspaces in the Polydisc

Year 2021, Volume: 16 Issue: 1, 121 - 128, 27.05.2021
https://doi.org/10.29233/sdufeffd.806065

Abstract

It is known from A. Beurling’s work [2] that any invariant subspace of the Hardy space H2(D) on the unit disc D is singly-generated and generated by an inner function. In contrast to the unit disc case, the structure of invariant subspaces in the polydisc is much more complicated. The problem of classification or explicit description of all invariant subspaces in the polydisc given by W.Rudin in his book “Function Theory in Polydiscs” is still one of the most important open problems of operator theory and function theory of complex variables, although it has been extensively studied. In this study, we give a characterization for singly-generated invariant subspaces of Hardy space H2(Dn) on the polydisc Dn by using the Beurling-Lax-Halmos Theorem. Moreover, it is also obtained the characterization of Beurling-type invariant subspaces given in [12].

References

  • [1] O.P. Agrawal, D.N. Clark and R.G. Douglas, “Invariant subspaces in the polydisk,” Pacific J. Math., 121(1), 1–11, 1986.
  • [2] A. Beurling, “On two problems concerning linear transformations in Hilbert space,” Acta Math., 81, 17 pp., 1948.
  • [3] P.R. Halmos, A Hilbert space problem book, 2nd edition, Springer-Verlag, New York-Berlin, 1982.
  • [4] C.A. Jacewicz, “A nonprincipal invariant subspace of the Hardy space on the torus,” Proc. Amer. Math. Soc., 31, 127–129, 1972.
  • [5] B.B. Koca, “Two types of invariant subspaces in the polydisc,” Results Math, 71, 1297–1305, 2017.
  • [6] B.B. Koca and N. Sadik, “Invariant subspaces generated by a single function in the polydisk,” Math. Notes, 102(1-2), 193–197, 2017.
  • [7] V. Mandrekar, “The validity of Beurling theorems in polydiscs,” Proc. Amer. Math. Soc., 103, 145–148, 1988.
  • [8] Y, Qin, R. Yang, “A characterization of submodules via Beurling-Lax-Halmos theorem,” Proc. Amer. Math. Soc, 142, 3505–3510, 2014.
  • [9] H. Radjavi and P. Rosenthal, Invariant subspaces, Springer-Verlag, New York-Heidelberg, 1973.
  • [10] J. Radlow, “Ideals of square summable power series in several variables,” Proc. Amer. Math. Soc., 38(2), 293–297, 1973.
  • [11] W. Rudin, Function theory in polydiscs, W. A. Benjamin, Inc., New York- Amsterdam, 1969.
  • [12] N.M. Sadikov, “Invariant subspaces in the Hardy space on a polydisk that are generated by inner functions,” Akad. Nauk Azerbaidzhan. SSR Dokl., 39 (3), 8–11, 1983 (Russian).
  • [13] J. Sarkar, “Submodules of the Hardy module over polydisc,” Israel Journal of Mathematics. 205, 317–336, 2015.
  • [14] J. Sarkar, A. Sasane and B. Wick, “Doubly commuting submodules of the Hardy module over polydiscs,” Studia Mathematica, 217(2), 179–192, 2013.
  • [15] M. Seto and R. Yang, “Inner sequence based invariant subspaces in H^2 (D^2 ),” Proc. Amer. Math. Soc., 135, 2519–2526, 2007.
  • [16] R. Yang, “Beurling’s phenomenon in two variables,” Integr. Equ. Oper. Theory, 48, 411–423, 2004.
  • [17] Y. Yang, “Two inner sequences based invariant subspaces in H^2 (D^2 ),” Integr. Equ. Oper. Theory., 77, 279–290, 2013.

Polidiskte Tek Bir Eleman Tarafından Üretilen Değişmez Alt Uzaylar için Operatör Teorik Bir Kuruluş

Year 2021, Volume: 16 Issue: 1, 121 - 128, 27.05.2021
https://doi.org/10.29233/sdufeffd.806065

Abstract

Birim disk D üzerindeki H2(D) Hardy uzayının değişmez alt uzaylarının tek bir fonksiyon tarafından üretilen alt uzaylar olduğu ve bu fonksiyonların bir iç fonksiyon olduğu A. Beurling’in [2] çalışmasından bilinmektedir. Birim diskteki bu durumun aksine, polidisk durumunda değişmez alt uzayların yapısı daha karmaşıktır. Öyle ki W. Rudin’in “Function Theory in Polydiscs” kitabında verdiği “Polidisk üzerindeki değişmez alt uzayların sınıflandırılması ya da kesin bir tanımının verilmesi” problemi yoğun bir şekilde çalışılmasına rağmen bugün hala operatör teori ve kompleks değişkenli fonksiyonlar teorisinin en önemli açık problemlerindendir. Bu çalışmada, Beurling-Lax-Halmos Teoremi kullanılarak, polidisk Dn üzerindeki H2(D)(Dn) Hardy uzayının tek bir eleman tarafından üretilen değişmez alt uzayları için bir karakterizasyon verilecektir. Ek olarak, Beurling-tipli değişmez alt uzaylar için [12]’de verilen bir karakterizasyon da elde edilecektir.

References

  • [1] O.P. Agrawal, D.N. Clark and R.G. Douglas, “Invariant subspaces in the polydisk,” Pacific J. Math., 121(1), 1–11, 1986.
  • [2] A. Beurling, “On two problems concerning linear transformations in Hilbert space,” Acta Math., 81, 17 pp., 1948.
  • [3] P.R. Halmos, A Hilbert space problem book, 2nd edition, Springer-Verlag, New York-Berlin, 1982.
  • [4] C.A. Jacewicz, “A nonprincipal invariant subspace of the Hardy space on the torus,” Proc. Amer. Math. Soc., 31, 127–129, 1972.
  • [5] B.B. Koca, “Two types of invariant subspaces in the polydisc,” Results Math, 71, 1297–1305, 2017.
  • [6] B.B. Koca and N. Sadik, “Invariant subspaces generated by a single function in the polydisk,” Math. Notes, 102(1-2), 193–197, 2017.
  • [7] V. Mandrekar, “The validity of Beurling theorems in polydiscs,” Proc. Amer. Math. Soc., 103, 145–148, 1988.
  • [8] Y, Qin, R. Yang, “A characterization of submodules via Beurling-Lax-Halmos theorem,” Proc. Amer. Math. Soc, 142, 3505–3510, 2014.
  • [9] H. Radjavi and P. Rosenthal, Invariant subspaces, Springer-Verlag, New York-Heidelberg, 1973.
  • [10] J. Radlow, “Ideals of square summable power series in several variables,” Proc. Amer. Math. Soc., 38(2), 293–297, 1973.
  • [11] W. Rudin, Function theory in polydiscs, W. A. Benjamin, Inc., New York- Amsterdam, 1969.
  • [12] N.M. Sadikov, “Invariant subspaces in the Hardy space on a polydisk that are generated by inner functions,” Akad. Nauk Azerbaidzhan. SSR Dokl., 39 (3), 8–11, 1983 (Russian).
  • [13] J. Sarkar, “Submodules of the Hardy module over polydisc,” Israel Journal of Mathematics. 205, 317–336, 2015.
  • [14] J. Sarkar, A. Sasane and B. Wick, “Doubly commuting submodules of the Hardy module over polydiscs,” Studia Mathematica, 217(2), 179–192, 2013.
  • [15] M. Seto and R. Yang, “Inner sequence based invariant subspaces in H^2 (D^2 ),” Proc. Amer. Math. Soc., 135, 2519–2526, 2007.
  • [16] R. Yang, “Beurling’s phenomenon in two variables,” Integr. Equ. Oper. Theory, 48, 411–423, 2004.
  • [17] Y. Yang, “Two inner sequences based invariant subspaces in H^2 (D^2 ),” Integr. Equ. Oper. Theory., 77, 279–290, 2013.
There are 17 citations in total.

Details

Primary Language Turkish
Subjects Mathematical Sciences
Journal Section Makaleler
Authors

Beyaz Başak Eskişehirli 0000-0002-6481-6020

Publication Date May 27, 2021
Published in Issue Year 2021 Volume: 16 Issue: 1

Cite

IEEE B. B. Eskişehirli, “Polidiskte Tek Bir Eleman Tarafından Üretilen Değişmez Alt Uzaylar için Operatör Teorik Bir Kuruluş”, Süleyman Demirel University Faculty of Arts and Science Journal of Science, vol. 16, no. 1, pp. 121–128, 2021, doi: 10.29233/sdufeffd.806065.