×

zbMATH — the first resource for mathematics

Spectral picture for rationally multicyclic subnormal operators. (English) Zbl 07002036
Summary: 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 \(\mathcal{K}_{m}=\operatorname{clos}(\operatorname{span}\{(M^{*})^{k}x:x\in\mathcal{H},0\leq k\leq m\}\). Suppose that \(M|_{\mathcal{K}_{N-1}}\) is pure. Then \(\operatorname{clos}(\sigma(S)\setminus\sigma_{e}(S))=\operatorname{clos}(\operatorname{Int}(\sigma(S)))\).
MSC:
47B20 Subnormal operators, hyponormal operators, etc.
47A16 Cyclic vectors, hypercyclic and chaotic operators
30H99 Spaces and algebras of analytic functions of one complex variable
PDF BibTeX XML Cite
Full Text: DOI arXiv Euclid
References:
[1] A. Aleman, S. Richter, and C. Sundberg, Nontangential limits in \(P^{t}(μ)\)-spaces and the index of invariant subspaces, Ann. of Math. (2) 169 (2009), no. 2, 449–490. · Zbl 1179.46020
[2] A. Aleman, S. Richter, and C. Sundberg, “A quantitative estimate for bounded point evaluations in \(P^{t}(μ)\)-spaces” in Topics in Operator Theory, Vol. 1: Operators, Matrices and Analytic Functions, Oper. Theory Adv. Appl. 202, Birkhäuser, Basel, 2010, 1–10. · Zbl 1201.47020
[3] J. E. Brennan, Point evaluations, invariant subspaces and approximation in the mean by polynomials, J. Funct. Anal. 34 (1979), no. 3, 407–420. · Zbl 0428.41005
[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. · Zbl 1161.30026
[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. · Zbl 0469.47020
[8] J. B. Conway, The Theory of Subnormal Operators, Math. Surveys Monogr. 36, Amer. Math. Soc. Providence, 1991. · Zbl 0743.47012
[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. · Zbl 0814.46038
[10] M. J. Cowen and R. G. Douglas, Complex geometry and operator theory, Acta. Math. 141 (1978), no. 3–4, 187–261. · Zbl 0427.47016
[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. · Zbl 1058.47018
[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. · Zbl 1448.47017
[15] J. E. McCarthy and L. Yang, Subnormal operators and quadrature domains, Adv. Math. 127 (1997), no. 1, 52–72. · Zbl 0902.47024
[16] P. McGuire, On the spectral picture of an irreducible subnormal operator, Proc. Amer. Math. Soc. 104 (1988), no. 3, 801–808. · Zbl 0693.47020
[17] R. F. Olin and J. E. Thomson, Irreducible operators whose spectra are spectral sets, Pacific J. Math. 91 (1980), no. 2, 431–434. · Zbl 0415.47003
[18] J. E. Thomson, Approximation in the mean by polynomials, Ann. of Math. (2) 133 (1991), no. 3, 477–507. · Zbl 0736.41008
[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. · Zbl 0838.47017
[20] D. V. Yakubovich, Subnormal operators of finite type, II: Structure theorems, Rev. Mat. Iberoam. 14 (1998), no. 3, 623–681. · Zbl 0933.47016
[21] L. Yang, A note on \(L^{p}\)-bounded point evaluations for polynomials, Proc. Amer. Math. Soc. 144 (2016), no. 11, 4943–4948. · Zbl 1345.30043
[22] L. Yang, Bounded point evaluations for rationally multicyclic subnormal operators, J. Math. Anal. Appl. 458 (2018), no. 2, 1059–1072. · Zbl 06813439
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.