Sectorial perturbations of self-adjoint matrices and operators

This paper considers N x N matrices of the form A(gamma)=A+gamma B, where A is self-adjoint, gamma is an element of C and B is a non-self-adjoint perturbation of A. We obtain some monodromy-type results relating the spectral behaviour of such matrices in the two asymptotic regimes vertical bar gamma vertical bar -> infinity and vertical bar gamma vertical bar -> 0 under certain assumptions on B. We also explain some properties of the spectrum of A(gamma) for intermediate-sized gamma by considering the limit N -> infinity, concentrating on properties that have no self-adjoint analogue. A substantial number of results extend to operators on infinite-dimensional Hilbert spaces.