  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 $T^*$
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.
