The topic of this diploma thesis is the computation of the spectrum and resolvent of bounded linear operators on a complex Banach space. We start by describing a method introduced by O. Nevanlinna for computing the spectrum and representing the resolvent. The method produces polynomial sublevel sets that converge to the polynomially convex hull of the spectrum. As a by-product, it also provides us with explicit expressions for the resolvent operator everywhere outside the obtained sets.

Nevanlinna's method is based on the computation of the ideal Arnoldi polynomials, and in the latter part of this thesis, we concentrate on studying the behavior of the polynomial lemniscates (level sets) of these polynomials. We consider two example cases. First, we discuss bounded linear operators on a sequence space !q(Z) and, after that, finite dimensional matrices. In the latter case, we also discuss some practical implementation problems faced when computing the polynomial lemniscates. Overall, our goal is to convince the reader, largely with the aid of pictures, that the polynomial lemniscates of ideal Arnoldi polynomials provide a useful tool for computing the spectrum and representing the resolvent.