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.