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On the convex hull generated by orbit of operators. (English) Zbl 1311.47012
In this paper, the new concept of convex-cyclicity, located between cyclicity and hypercyclicity, is studied. A bounded linear operator $$T$$ on a Banach space $$X$$ is said to be convex-cyclic if the convex hull generated by the orbit of some $$x \in X$$ under $$T$$ is dense in $$X$$. Convex-cyclicity can also be considered on finite dimensional vector spaces. Using the Jordan decomposition theorem, the author has characterized all convex-cyclic operators on finite dimensional vector spaces.
Furthermore, a comparison between hypercyclicity and convex-cyclicity is carried out by the author. The main results of the paper, in this connection, run as follows.
Let $$T^*$$ be the adjoint of $$T$$, $$\sigma(T)$$ and $$\sigma_p(T)$$ stand, respectively, for the spectrum and the point spectrum of $$T$$, and $$\mathbb{D}$$ be the open unit disc.
{1.} A necessary condition for the convex-cyclicity of an operator on a complex Banach space is that $$\sigma_p(T^*)$$ does not intersect $$\mathbb{D}$$ and the real line $$\mathbb{R}$$; nonetheless, on a real Banach space, it does not intersect the interval $$[-1,+\infty)$$.
{2.} Similar to hypercyclic operators, if $$T$$ is convex-cyclic and $$\sigma_p(T^*)=\varnothing$$, then $$T$$ has an invariant dense subspace of convex-cyclic vectors.
{3.} It is known that every component of the spectrum of a hypercyclic operator intersects the boundary of $$\mathbb{D}$$. A technique similar to the proof of this fact is used to show that each component of the spectrum of a convex-cyclic operator intersects $$\mathbb{C}\backslash \mathbb{D}$$.
{4.} The convex-transitivity of an operator $$T$$ is defined. It implies the convex-cyclicity of $$T$$; the converse is true if $$\sigma_p(T^*)=\varnothing$$.
{5.} A criterion, similar to the hypercyclicity criterion, is presented that ensures a linear operator to be convex-cyclic. It is used to give a sufficient condition for the convex-cyclicity of a bilateral weighted backward shift operator acting on $$l^p(\mathbb{Z})$$, $$1\leq p <\infty$$. However, for the backward shifts acting on $$l^p(\mathbb{N})$$, convex-cyclicity and hypercyclicity coincide.
{6.} It is known that the inverse of a hypercyclic operator is hypercyclic. In spite of this fact, the inverse of a convex-cyclic operator need not be convex-cyclic.
{7.} The hypercyclicity of composition operators has been studied in the literature. The problem of their convex-cyclicity is considered in this paper. It is proved that the composition operator induced by a parabolic non-authomorphism self-map of $$\mathbb{D}$$ on the classical Hardy space $$H^2$$ is not convex-cyclic.

MSC:
 47A16 Cyclic vectors, hypercyclic and chaotic operators 52A07 Convex sets in topological vector spaces (aspects of convex geometry)
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References:
 [1] Ansari, S. I., Hypercyclic and cyclic vectors, J. Funct. Anal., 128, 374-383, (1995) · Zbl 0853.47013 [2] Bayart, F.; Matheron, E., Dynamics of linear operators, (2009), Cambridge University Press New York · Zbl 1187.47001 [3] Bes, J., Invariant manifolds of hypercyclic vectors for the real scalar case, Proc. Amer. Math. Soc., 127, 1801-1804, (1999) · Zbl 0914.47005 [4] Bes, J.; Peris, A., Hereditarily hypercyclic operators, J. Funct. Anal., 167, 94-112, (1999) · Zbl 0941.47002 [5] Birkhoff, G. D., Demonstration d’un theoreme elementaire sur LES fonctions entieres, C. R. Acad. Sci. Paris, 189, 473-475, (1929) · JFM 55.0192.07 [6] Bourdon, P. S., Invariant manifolds of hypercyclic vectors, Proc. Amer. Math. Soc., 118, 845-847, (1993) · Zbl 0809.47005 [7] Bourdon, P. S.; Feldman, N. S., Somewhere dense orbits are everywhere dense, Indiana Univ. Math. J., 52, 811-819, (2003) · Zbl 1049.47002 [8] Bourdon, P. S.; Shapiro, J. H., Cyclic phenomena for composition operators, Mem. Amer. Math. Soc., 596, (1997) · Zbl 0996.47032 [9] Cowen, C. C., Linear fractional composition operators on $$H^2$$, Integral Equations Operator Theory, 11, 151-160, (1988) · Zbl 0638.47027 [10] Cowen, C. C.; MacCluer, B., Composition operators on spaces of analytic functions, studies in advanced mathematics, (1995), CRC Press · Zbl 0873.47017 [11] Gethner, R. M.; Shapiro, J. H., Universal vectors for operators on spaces of holomorphic functions, Proc. Amer. Math. Soc., 100, 2, 281-288, (1987) · Zbl 0618.30031 [12] Godefroy, G.; Shapiro, J., Operators with dense, invariant, cyclic vector manifolds, J. Funct. Anal., 98, 229-269, (1991) · Zbl 0732.47016 [13] Grosse-Erdmann, K. G.; Peris, A., Linear chaos, universitext, (2011), Springer [14] Herrero, D. A., Limits of hypercyclic and supercyclic operators, J. Funct. Anal., 99, 179-190, (1991) · Zbl 0758.47016 [15] Hilden, H. M.; Wallen, L. J., Some cyclic and non-cyclic vectors of certain operators, Indiana Univ. Math. J., 24, 557-565, (1974) · Zbl 0274.47004 [16] C. Kitai, Invariant closed sets for linear operators, Ph.D. Thesis, University of Toronto, 1982. [17] Rolewicz, S., On orbit elements, Studia Math., 32, 17-22, (1969) · Zbl 0174.44203 [18] Salas, H., Hypercyclic weighted shifts, Trans. Amer. Math. Soc., 347, 993-1004, (1995) · Zbl 0822.47030
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