Banach spaces valued ergodic theorems and spectral approximation The present dissertation thesis is concerned with Banach space valued convergence theorems along Folner type sequences in geometries with amenable structures. The text contains new ergodic theorems for groups as well as an almost additive and a subadditive convergence theorem for hyperfinite weakly convergent graph sequences. All these results among them abstract mean and pointwise ergodic theorems significantly extend the literature on ergodic theorems. As an application one obtains the uniform convergence of the integrated density of states (IDS) for pattern invariant finite hopping range operators in a wide range of amenable situations. A different kind of spectral approximation is given by a strong result for Iharas Zeta function associated with sofic graphings. It is shown that compact convergence can be obtained for the normalized finite versions corresponding to the elements in a weakly convergent graph sequence.