In this thesis, various aspects of finite - not necessarily commuting - tuples of continuous linear operators on Banach spaces or Hilbert spaces are considered. We derive generalizations and analogues of some results which are well known for single operators and commuting tuples.
In the first part we study the representation theory for such non-commuting tuples. To be more precisely, we consider multiplication operators between Fock spaces and prove a certain Commutant-Lifting theorem. Moreover, the characteristic function of an n-contraction is analyzed.
The main focus in the second part is put on semi-Fredholm tuples. We introduce the Hilbert-Samuel polynomial for such tuples and compute its degree and leading coefficient. Furthermore, we prove that the range of certain semi-Fredholm functions is closed in order to analyze some spectral subspaces, which are well known in the commutative local spectral theory.