Subspace-hypercyclic weighted shifts

Abstract

Our aim in this paper is to obtain necessary and sufficient conditions for weighted shift operators on the Hilbert spaces ${\ell }^2(\mathbb{Z})$ and ${\ell }^2(\mathbb{N})$ to be subspace-transitive, consequently, we show that the Herrero question (D. A. Herrero. Limits of hypercyclic and supercyclic operators, J. Funct. Anal., 99 (1991)179-190) holds true even on a subspace of a Hilbert space, i.e. there exists an operator $T$ such that both $T$ and $T^{\ast }$ are subspace-hypercyclic operators for some subspaces. We display the conditions on the direct sum of two invertable bilateral forward weighted shift operators to be subspace-hypercyclic.