Orbits of operators and operator semigroups

Denote byB(X) the set of all bounded linear operators acting on a Banach spaceX . For simplicity we assume that all Banach spaces are complex unless stated explicitly otherwise. However, the notions make sense also in real Banach spaces and most of the results remain true (with slight modifications) in the real case. Let X be a Banach space and let T be a bounded linear operator on X . By an orbit of T we mean a sequence of the form (Tx)n=0, where x ∈ X is a fixed vector. By a weak orbit of T we mean a sequence of the form (〈Tnx, x〉)n=0, where x ∈ X and x are fixed vectors. Orbits of operators appear frequently in operator theory. They are closely connected with the famous invariant subspace/subset problem, they play a central role in the linear dynamics and appear also in other branches of operator theory, for example in the local spectral theory or in the theory of operator semigroup. Typically, the behaviour of an orbit (Tx) depends essentially on the choice of the initial vector x. This can be illustrated by the following simple example: