On subspace convex-cyclic operators

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.