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International Journal of Mathematics and Mathematical Sciences
Volume 2009, Article ID 239025, 17 pages
Research Article

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

1Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409, USA
2Institute for Numerical and Applied Mathematics, University of Göttingen, 37083 Göttingen, Germany
3Department of Mathematics, Utah Valley University, Orem, UT 84058, USA

Received 7 June 2008; Accepted 15 January 2009

Academic Editor: Petru Jebelean

Copyright © 2009 D. S. Gilliam et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.