Infinite-dimensional features of matrices and pseudospectra

Abstract

Abstract. Given a Hilbert space operator T , the level sets of function Ψ T (z) = (T − z) −1 −1 determine the so-called pseudospectra of T . We set Ψ T to be zero on the spectrum of T . After giving some elementary properties of Ψ T (which, as it seems, were not noticed before), we apply them to the study of the approximation. We prove that for any operator T , there is a sequence {T n } of finite matrices such that Ψ Tn (z) tends to Ψ T (z) uniformly on C. In this proof, quasitriangular operators play a special role. This is merely an existence result, we do not give a concrete construction of this sequence of matrices.One of our main points is to show how to use infinite-dimensional operator models in order to produce examples and counterexamples in the set of finite matrices of large size. In particular, we get a result, which means, in a sense, that the pseudospectrum of a nilpotent matrix can be anything one can imagine. We also study the norms of the multipliers in the context of Cowen-Douglas class operators. We use these results to show that, to the opposite to the function Ψ S , the function √ S − z for certain finite matrices S may oscillate arbitrarily fast even far away from the spectrum.