The Fréchet Derivative of an Analytic Function of a Bounded Operator with Some Applications

The main result in this paper is the determination of the Fréchet derivative of an analytic function of a bounded operator, tangentially to the space of all bounded operators. Some applied problems from statistics and numerical analysis are included as a motivation for this study. The perturbation operator (increment) is not of any special form and is not supposed to commute with the operator at which the derivative is evaluated. This generality is important for the applications. In the Hermitian case, moreover, some results on perturbation of an isolated eigenvalue, its eigenprojection, and its eigenvector if the eigenvalue is simple, are also included. Although these results are known in principle, they are not in general formulated in terms of arbitrary perturbations as required for the applications. Moreover, these results are presented as corollaries to the main theorem, so that this paper also provides a short, essentially self-contained review of these aspects of perturbation theory.