Abstract

During the last years pseudospectra have become increasingly popular and have lead to interesting results especially for operators on Hilbert spaces. In the more general context of Banach spaces additional difficulties occur, among them the fact, which was observed by Shargorodsky, that the robustness of pseudospectra under perturbations can get lost. In the first part of this talk I want to discuss how the recently introduced concept of (N,epsilon)-pseudospectra permits to overcome this drawback in a sense and to extend a couple of interesting results beyond the Hilbert space world. In the second part we pick up the well known observation that passing from an operator A to its finite sections can cause additional components in the respective pseudospectra. I want to present an algebraic framework which can provide an explanation for this phenomenon for several classes of operators.