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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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