Eigenelements of perturbed operators

Abstract

Let lambda(0) be a semisimple eigenvalue of an operator T(0). Let gamma(0) be a circle with centre lambda s(0) containing no other spectral value of T(0). Some lower bounds are obtained for the convergence radius of the power series for the spectral projection P(t) and for trace T(t)P(t) associated with linear perturbation family T(t) = T(0) + tV(0) and the circle gamma(0). They are useful when T(0) is a member of a sequence (Tn) which approximates an operator T in a collectively compact manner. These bounds result from a modification of Kato's method of majorizing series, based on an idea of Redont. I lambda(0) is simple, it is shown that the same lower bound are valid for the convergence radius of a power series yielding an eigenvector of T(t).