Journal of Mathematical Physics, Analysis, Geometry 2020, vol. 16, No 4, pp. 473-489   https://doi.org/10.15407/mag16.04.473     ( to contents , go back )

### On Subspace Convex-Cyclic Operators

Jarosław Woźniak

Institute of Mathematics, Department of Mathematics and Physics, University of Szczecin, ul. Wielkopolska 15, 70-451 Szczecin, Poland
E-mail: wozniak@univ.szczecin.pl

Dilan Ahmed

University of Sulaimani, College of Education, Department of Mathematics, Kurdistan Region, Sulaimani, Iraq
Komar University of Science and Technology, Computer Engineering Department, Kurdistan Region, Sulaimani, Iraq
E-mail: dilan.ahmed@univsul.edu.iq

Mudhafar Hama

University of Sulaimani, College of Science, Department of Mathematics, Kurdistan Region, Sulaimani, Iraq
E-mail: mudhafar.hama@univsul.edu.iq

Karwan Jwamer

University of Sulaimani, College of Science, Department of Mathematics, Kurdistan Region, Sulaimani, Iraq
E-mail: :karwan.jwamer@univsul.edu.iq

Received October 9, 2019, revised May 12, 2020.

Abstract

Let $\mathcal{H}$ be an infinite dimensional real or complex separable Hilbert space. We introduce a special type of a bounded linear operator $T$ and study its important relation with the invariant subspace problem on $\mathcal{H}$: the operator $T$ is said to be subspace convex-cyclic for a subspace $\mathcal{M}$ if there exists a vector whose orbit under $T$ intersects the subspace $\mathcal{M}$ in a relatively dense set. We give the sufficient condition for a subspace convex-cyclic transitive operator $T$ to be subspace convex-cyclic. We also give a special type of the Kitai criterion related to invariant subspaces which implies subspace convex-cyclicity. Finally we show a counterexample of a subspace convex-cyclic operator which is not subspace convex-cyclic transitive.

Mathematics Subject Classification 2010: 47A16, 37A25
Key words: ergodic dynamical systems, convex-cyclic operators, Kitai criterion, convex-cyclic transitive operators