×

Limits of hypercyclic and supercyclic operators. (English) Zbl 0758.47016

Let \(T\) be a bounded linear operator on a complex, separable infinite dimensional Hilbert space \(H\), and let \(y\in H\). The orbit of \(y\) under \(T\) is the set \(\text{Orb}(T,y):=\{y,Ty,T^ 2y,\dots\}\). A vector \(y\in H\) is called hypercyclic (resp., supercyclic) for \(T\), if \(\text{Orb}(T,y)\) is dense in \(H\) (resp., if the set of scalar multiples of \(\text{Orb}(T,y)\) is dense in \(H\)). The author obtains a spectral characterization of the norm closure of the class of all hypercyclic operators on \(H\), and describes the structure of the set of all hypercyclic vectors of a given hypercyclic operator. Analogous results are obtained for supercyclic operators and vectors.

MSC:

47A65 Structure theory of linear operators
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Apostol, C., The correction by compact perturbations of the singular behavior of operators, Rev. Roumaine Math. Pures Appl., 21, 155-175 (1976) · Zbl 0336.47012
[2] Apostol, C.; Fialkow, L. A.; Herrero, D. A.; Voiculescu, D., Approximation of Hilbert space operators, I, (Research Notes in Math., Vol. 102 (1984), Pitman: Pitman Boston/London/Melbourne) · Zbl 0572.47001
[3] Beauzamy, B., Introduction to Operator Theory and Invariant Subspaces (1988), North-Holland: North-Holland Amsterdam/New York/Oxford/Tokyo · Zbl 0663.47002
[5] Gehtner, R. M.; Shapiro, J. H., Universal vectors for operators on spaces of holomorphic functions, (Proc. Amer. Math. Soc., 100 (1987)), 281-288 · Zbl 0618.30031
[7] Herrero, D. A., Approximation of Hilbert space operators, (Pitman Research Notes in Math. Series, Vol. 224 (1989), Longman Scientific and Technical: Longman Scientific and Technical Harlow, Essex, England), Wiley, New York · Zbl 0408.47001
[8] Herrero, D. A., The distance to a similarity-invariant set of operators, Integral Equations Operator Theory, 5, 131-140 (1982) · Zbl 0499.47012
[9] Herrero, D. A., Economical compact perturbations, I, Erasing normal eigenvalues, J. Operator Theory, 10, 289-306 (1983) · Zbl 0543.47013
[10] Herrero, D. A., On multicyclic operators, II, Two extensions of the notion of quasitriangularity, (Proc. London Math. Soc. (3), 48 (1984)), 247-282 · Zbl 0554.47004
[11] Herrero, D. A., The diagonal entries in the formula “quasitriangular — compact = triangular,” and restrictions of quasitriangularity, Trans. Amer. Math. Soc., 298, 1-42 (1986) · Zbl 0614.47014
[12] Herrero, D. A., Spectral pictures of operators in the Cowen-Douglas class \(B_n\)(Ω) and its closure, J. Operator Theory, 8, 213-222 (1987) · Zbl 0649.47013
[13] Herrero, D. A., Economical compact perturbations, II, Filling in the holes, J. Operator Theory, 19, 25-42 (1988) · Zbl 0673.47014
[14] Herrero, D. A., Similarity and Approximation of Operators, (Proceedings, Symposia in Pure Math.. Proceedings, Symposia in Pure Math., Amer Math. Soc., Providence, RI (July 1988), AMS Summer Institute on Operator Theory/Operator Algebras and Applications, Univ. of New Hampshire), to appear · Zbl 0408.47001
[16] Herrero, D. A., All (all?) about triangular operators (1989), preprint
[17] Herrero, D. A.; Wang, Z.-Y, Compact perturbations of hypercyclic and supercyclic operators (1989), preprint
[18] Kitai, C., Invariant Closed Sets for Linear Operators, (Thesis (1982), Univ. of Toronto)
[19] Riesz, F.; Sz.-Nagy, B., Functional Analysis (1955), Ungar: Ungar New York
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.