Banach Journal of Mathematical Analysis

Spectral picture for rationally multicyclic subnormal operators

Liming Yang

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

For a pure bounded rationally cyclic subnormal operator S on a separable complex Hilbert space H, Conway and Elias showed that clos(σ(S)σe(S))=clos(Int(σ(S))). This article examines the property for rationally multicyclic (N-cyclic) subnormal operators. We show that there exists a 2-cyclic irreducible subnormal operator S with clos(σ(S)σe(S))clos(Int(σ(S))). We also show the following. For a pure rationally N-cyclic subnormal operator S on H with the minimal normal extension M on KH, let Km=clos(span{(M)kx:xH,0km}. Suppose that M|KN1 is pure. Then clos(σ(S)σe(S))=clos(Int(σ(S))).

Article information

Source
Banach J. Math. Anal., Volume 13, Number 1 (2019), 151-173.

Dates
Received: 14 March 2018
Accepted: 21 June 2018
First available in Project Euclid: 28 September 2018

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1538121809

Digital Object Identifier
doi:10.1215/17358787-2018-0020

Mathematical Reviews number (MathSciNet)
MR3894066

Zentralblatt MATH identifier
07002036

Subjects
Primary: 47B20: Subnormal operators, hyponormal operators, etc.
Secondary: 47A16: Cyclic vectors, hypercyclic and chaotic operators 30H99: None of the above, but in this section

Keywords
spectral picture multicyclic subnormal operators

Citation

Yang, Liming. Spectral picture for rationally multicyclic subnormal operators. Banach J. Math. Anal. 13 (2019), no. 1, 151--173. doi:10.1215/17358787-2018-0020. https://projecteuclid.org/euclid.bjma/1538121809


Export citation

References

  • [1] A. Aleman, S. Richter, and C. Sundberg, Nontangential limits in $P^{t}(\mu)$-spaces and the index of invariant subspaces, Ann. of Math. (2) 169 (2009), no. 2, 449–490.
  • [2] A. Aleman, S. Richter, and C. Sundberg, “A quantitative estimate for bounded point evaluations in $P^{t}(\mu)$-spaces” in Topics in Operator Theory, Vol. 1: Operators, Matrices and Analytic Functions, Oper. Theory Adv. Appl. 202, Birkhäuser, Basel, 2010, 1–10.
  • [3] J. E. Brennan, Point evaluations, invariant subspaces and approximation in the mean by polynomials, J. Funct. Anal. 34 (1979), no. 3, 407–420.
  • [4] J. E. Brennan, Thomson’s theorem on mean-square polynomial approximation (in Russian), Algebra i Analiz 17, no. 2 (2005), 1–32; English translation in St. Petersburg Math. J. 17 (2006), 217–238.
  • [5] J. E. Brennan, The structure of certain spaces of analytic functions, Comput. Methods Funct. Theory 8 (2008), no. 1–2, 625–640.
  • [6] J. E. Brennan and E. R. Militzer, $L^{p}$-bounded point evaluations for polynomials and uniform rational approximation (in Russian), Algebra i Analiz 22, no. 1 (2010), 57–74; English translation in St. Petersburg Math. J. 22 (2011), 41–53.
  • [7] J. B. Conway, The dual of a subnormal operator, J. Operator Theory 5 (1981), no. 2, 195–211.
  • [8] J. B. Conway, The Theory of Subnormal Operators, Math. Surveys Monogr. 36, Amer. Math. Soc. Providence, 1991.
  • [9] J. B. Conway and N. Elias, Analytic bounded point evaluations for spaces of rational functions, J. Funct. Anal. 117 (1993), no. 1, 1–24.
  • [10] M. J. Cowen and R. G. Douglas, Complex geometry and operator theory, Acta. Math. 141 (1978), no. 3–4, 187–261.
  • [11] N. S. Feldman and P. McGuire, On the spectral picture of an irreducible subnormal operator, II, Proc. Amer. Math. Soc. 131 (2003), no. 6, 1793–1801.
  • [12] T. W. Gamelin, Uniform Algebras, Prentice-Hall Ser. Modern Anal., Prentice-Hall, Englewood Ciffs, N.J., 1969.
  • [13] S. Hruscev, The Brennan alternative for measures with finite entropy (in Russian), Izv. Akad. Nauk Armyan. SSR Ser. Math. 14 (1979), no. 3, 184–191.
  • [14] M. Mbekhta, N. Ourchane, and E. H. Zerouali, The interior of bounded point evaluations for rationally cyclic operators, Mediterr. J. Math. 13 (2016), no. 4, 1981–1996.
  • [15] J. E. McCarthy and L. Yang, Subnormal operators and quadrature domains, Adv. Math. 127 (1997), no. 1, 52–72.
  • [16] P. McGuire, On the spectral picture of an irreducible subnormal operator, Proc. Amer. Math. Soc. 104 (1988), no. 3, 801–808.
  • [17] R. F. Olin and J. E. Thomson, Irreducible operators whose spectra are spectral sets, Pacific J. Math. 91 (1980), no. 2, 431–434.
  • [18] J. E. Thomson, Approximation in the mean by polynomials, Ann. of Math. (2) 133 (1991), no. 3, 477–507.
  • [19] D. Xia, On pure subnormal operators with finite rank self-commutators and related operator tuples, Integral Equations Operator Theory 24 (1996), no. 1, 106–125.
  • [20] D. V. Yakubovich, Subnormal operators of finite type, II: Structure theorems, Rev. Mat. Iberoam. 14 (1998), no. 3, 623–681.
  • [21] L. Yang, A note on $L^{p}$-bounded point evaluations for polynomials, Proc. Amer. Math. Soc. 144 (2016), no. 11, 4943–4948.
  • [22] L. Yang, Bounded point evaluations for rationally multicyclic subnormal operators, J. Math. Anal. Appl. 458 (2018), no. 2, 1059–1072.