Operators with dense, invariant, cyclic vector manifolds. (English) Zbl 0732.47016

Let X be a Banach space, T a bounded operator on X, and \(x\in X\). The vector x is hypercyclic for T if the set \({\mathcal O}_ x=\{x,Tx,T^ 2x,...\}\) is dense in X; x is supercyclic if \(\cup_{\lambda \in {\mathbb{C}}}\lambda {\mathcal O}_ x\) is dense in X; x is cyclic if the linear space generated by \({\mathcal O}_ x\) is dense in X. The phenomenon of hypercyclicity was regarded as a somewhat unusual thing, and few examples of hypercyclic vectors were known (see the references in this article). The authors give a general study of the phenomenon, and construct remarkable classes of operators which have numerous hypercyclic vectors. For instance, they show that there are certain operators S such that every operator \(T\neq 0\) commuting with S has a dense linear manifold consisting entirely of hypercyclic vectors (except the origin). The methods used are quite elementary and very ingenious.
The authors point out that their results are not relevant for the invariant subspace problem since all of the operators they construct have many invariant subspaces. They point out to some connections between the material of the paper and the theory of chaotic transformations.


47A65 Structure theory of linear operators
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[1] Ahlfors, L. V., Complex Analysis (1966), McGraw-Hill: McGraw-Hill New York · Zbl 0154.31904
[2] Beauzamy, B., An operator on a separable Hilbert space, with many hypercyclic vectors, Studia Math., 87, 71-78 (1988) · Zbl 0647.46024
[3] Beauzamy, B., Un opérateur sur l’espace de Hilbert, dont tous les polynômes sont hypercycliques, C. R. Acad. Sci. Paris Ser. I, 303, 923-927 (1986) · Zbl 0612.47003
[5] Birkhoff, G. D., Démonstration d’un théorème élémentaire sur les fonctions entières, C. R. Acad. Sci. Paris, 189, 473-475 (1929)
[6] Bourdon, P.; Shapiro, J. H., Cyclic composition operators on \(H^2\), (Proc. Sympos. Pure Math., 51 (1990)), 43-53, Part 2 · Zbl 0729.47029
[8] Bourdon, P.; Shapiro, J. H., Spectral synthesis and common cyclic vectors, Michigan Math. J., 37, 71-90 (1990) · Zbl 0703.47025
[9] Chan, K. C., Common cyclic vectors for operator algebras on spaces of analytic functions, Indiana Univ. Math. J., 37, 919-928 (1988) · Zbl 0638.47035
[10] Chan, K. C., Common cyclic entire functions for partial differential operators, Integral Equations Operator Theory, 13, 132-137 (1990) · Zbl 0723.47044
[11] Clancey, K. F.; Rogers, D. D., Cyclic vectors and seminormal operators, Indiana Univ. Math. J., 27, 689-696 (1978) · Zbl 0396.47016
[12] Devaney, R. L., An Introduction to Chaotic Dynamical Systems (1989), Addison-Wesley: Addison-Wesley Reading, MA · Zbl 0695.58002
[13] Ehrenpreis, L., Mean periodic funçtions I, Amer. J. Math., 77, 293-328 (1955) · Zbl 0068.31702
[14] Enflo, P., On the invariant subspace problem for Banach spaces, Acta Math., 158, 213-313 (1987) · Zbl 0663.47003
[15] Fleming, R. J.; Jamison, J. E., Commutants of triangular matrix operators, Indiana Univ. Math. J., 28, 791-802 (1979) · Zbl 0396.47013
[16] Gethner, 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
[17] Halmos, P. R., (A Hilbert Space Problem Book (1982), Springer-Verlag: Springer-Verlag New York) · Zbl 0496.47001
[20] Hilden, H. M.; Wallen, L. J., Some cyclic and non-cyclic vectors of certain operators, Indiana Univ. Math. J., 23, 557-565 (1974) · Zbl 0274.47004
[21] Hormander, L., An Introduction to Complex Analysis in Several Variables (1966), Van Nostrand: Van Nostrand Princeton, NJ · Zbl 0138.06203
[22] Kitai, C., Invariant Closed Sets for Linear Operators, (Thesis (1982), University of Toronto)
[23] MacLane, G. R., Sequences of derivatives and normal families, J. Analyse Math., 2, 72-87 (1952) · Zbl 0049.05603
[24] Malgrange, B., Existence et approximation des solutions des équations aux dérivées partiellles et des équations de convolution, Ann. Inst. Fourier (Grenoble), 8, 271-355 (1955/1956) · Zbl 0071.09002
[25] Meise, R.; Taylor, B. A., Each non-zero convolution operator on the entire functions admits a continuous linear right inverse, Math. Z., 197, 139-152 (1988) · Zbl 0618.32014
[26] Nikolskii, N. K.; Vasunin, V. I., Control subspaces of minimal dimension and root vectors, Integral Equations Operator Theorem, 6, 274-311 (1983) · Zbl 0513.47002
[27] Read, C., A solution to the invariant subspace problem, Bull. London Math. Soc., 16, 337-401 (1984) · Zbl 0566.47003
[28] Read, C., A solution to the invariant subspace problem on the space \(l^1\), Bull. London Math. Soc., 17, 305-317 (1985) · Zbl 0574.47006
[29] Read, C., The invariant subspace problem for a class of Banach spaces, 2: Hypercyclic operators, Israel J. Math., 63, 1-40 (1988) · Zbl 0782.47002
[30] Rolewicz, S., On orbits of elements, Studia Math., 32, 17-22 (1969) · Zbl 0174.44203
[31] Rudin, W., Function Theory in Polydiscs (1969), Benjamin: Benjamin New York · Zbl 0177.34101
[32] Rudin, W., Function Theory in the Unit Ball of \(C^N (1980)\), Springer-Verlag: Springer-Verlag New York · Zbl 0495.32001
[33] Saks, S.; Zygmund, A., Analytic Functions (1965), Polish Scientific Publishers: Polish Scientific Publishers Warsaw · Zbl 0136.37301
[35] Schwartz, L., Théorie générale des fonctions moyenne-périodiques, Ann. of Math., 48, 857-929 (1947) · Zbl 0030.15004
[36] Shields, A. L., Weighted shift operators and analytic function theory, (Topics in Operator Theory. Topics in Operator Theory, Mathematical Surveys, Vol. 13 (1974), Amer. Math. Soc: Amer. Math. Soc Providence RI) · Zbl 0303.47021
[37] Shields, A. L.; Walen, L. J., The commutants of certain Hilbert space operators, Indiana Univ. Math. J., 20, 777-788 (1971) · Zbl 0195.13601
[38] Stegenga, D. A., Multipliers of the Dirichlet space, Illinois J. Math., 24, 113-139 (1980) · Zbl 0432.30016
[39] Wermer, J., On invariant subspaces of normal operators, (Proc. Amer. Math. Soc., 3 (1952)), 270-277 · Zbl 0046.33704
[40] Wogen, W. R., On some operators with cyclic vectors, Indiana Univ. Math. J., 27, 163-171 (1978) · Zbl 0347.47014
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