Operators similar to their restrictions to invariant subspaces

An operator T on a separable infinite dimensional Banach space X is called crystal-like if for each nonzero invariant subspace M of T, there exists a positive number r such that T|M, the restriction of T to M, is similar to rT. In this paper, we investigate some elementary properties of crystal-like operators, such as the spectrum, cyclicity, strong irreducibility, essential similarity, reflectivity and the commutativity of the commutant.