On the convex hull generated by orbit of operators.

*(English)*Zbl 1311.47012In 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.

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.

Reviewer: Masoumeh Faghih-Ahmadi (Shiraz)

##### MSC:

47A16 | Cyclic vectors, hypercyclic and chaotic operators |

52A07 | Convex sets in topological vector spaces (aspects of convex geometry) |

##### Keywords:

convex-cyclic operator; hypercyclic operator; convex hull; backward shift; composition operator
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\textit{H. Rezaei}, Linear Algebra Appl. 438, No. 11, 4190--4203 (2013; Zbl 1311.47012)

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