On Subspace Convex-Cyclic Operators

Abstract

Let 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 H: the operator T is said to be subspace convex-cyclic for a subspace M if there exists a vector whose orbit under T intersects the subspace 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 convexcyclic operator which is not subspace convex-cyclic transitive.