## Banach Journal of Mathematical Analysis

### Spectral picture for rationally multicyclic subnormal operators

Liming Yang

#### Abstract

For a pure bounded rationally cyclic subnormal operator $S$ on a separable complex Hilbert space $\mathcal{H}$, Conway and Elias showed that $\operatorname{clos}(\sigma(S)\setminus\sigma_{e}(S))=\operatorname{clos}(\operatorname{Int}(\sigma(S)))$. 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 $\operatorname{clos}(\sigma(S)\setminus\sigma_{e}(S))\neq\operatorname{clos}(\operatorname{Int}(\sigma(S)))$. We also show the following. For a pure rationally $N$-cyclic subnormal operator $S$ on $\mathcal{H}$ with the minimal normal extension $M$ on $\mathcal{K}\supset\mathcal{H}$, let $\mathcal{K}_{m}=\operatorname{clos}(\operatorname{span}\{(M^{*})^{k}x:x\in\mathcal{H},0\le k\le m\}$. Suppose that $M|_{\mathcal{K}_{N-1}}$ is pure. Then $\operatorname{clos}(\sigma(S)\setminus\sigma_{e}(S))=\operatorname{clos}(\operatorname{Int}(\sigma(S)))$.

#### Article information

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

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

https://projecteuclid.org/euclid.bjma/1538121809

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

Mathematical Reviews number (MathSciNet)
MR3894066

Zentralblatt MATH identifier
07002036

#### 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

#### 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.