ABSTRACT
We discuss a fairly general approach to proving the closedness and upper semicontinuity of the multivalued map which associates to each continuous nonlinear operator from a certain operator class a natural spectrum. In this way, we show that the Kachurovskij spectrum for Lip- schitz continuous operators, the Furi-Martelli-Vignoli spectrum for quasibounded operators, and the Feng spectrum for so-called fc-epi operators are all upper semicontinuous with respect to a suitable normed or locally convex topology. On the other hand, we give a simple counterexample which shows that the Dorfner spectrum for linearly bounded operators has not a closed graph and is not upper semicontinuous either.
