Orbits, Weak Orbits and Local Capacity of Operators

Let T be an operator on a Banach space X. We give a survey of results concerning orbits {Tnx : n = 0, 1, . . .} and weak orbits {〈Tnx, x∗〉 : n = 0, 1, . . .} of T where x ∈ X and x∗ ∈ X∗. Further we study the local capacity of operators and prove that there is a residual set of points x ∈ X with the property that the local capacity cap(T, x) is equal to the global capacity cap T . This is an analogy to the corresponding result for the local spectral radius.