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 o...
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| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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Wiley
2009-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/2009/239025 |
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| author | D. S. Gilliam T. Hohage X. Ji F. Ruymgaart |
| author_facet | D. S. Gilliam T. Hohage X. Ji F. Ruymgaart |
| author_sort | D. S. Gilliam |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-5cfa530d48f545c397fe52d23f386a79 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2009-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-5cfa530d48f545c397fe52d23f386a792025-02-03T05:57:41ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252009-01-01200910.1155/2009/239025239025The Fréchet Derivative of an Analytic Function of a Bounded Operator with Some ApplicationsD. S. Gilliam0T. Hohage1X. Ji2F. Ruymgaart3Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409, USAInstitute for Numerical and Applied Mathematics, University of Göttingen, 37083 Göttingen, GermanyDepartment of Mathematics, Utah Valley University, Orem, UT 84058, USADepartment of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409, USAThe 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.http://dx.doi.org/10.1155/2009/239025 |
| spellingShingle | D. S. Gilliam T. Hohage X. Ji F. Ruymgaart The Fréchet Derivative of an Analytic Function of a Bounded Operator with Some Applications International Journal of Mathematics and Mathematical Sciences |
| title | The Fréchet Derivative of an Analytic Function of a Bounded Operator with Some Applications |
| title_full | The Fréchet Derivative of an Analytic Function of a Bounded Operator with Some Applications |
| title_fullStr | The Fréchet Derivative of an Analytic Function of a Bounded Operator with Some Applications |
| title_full_unstemmed | The Fréchet Derivative of an Analytic Function of a Bounded Operator with Some Applications |
| title_short | The Fréchet Derivative of an Analytic Function of a Bounded Operator with Some Applications |
| title_sort | frechet derivative of an analytic function of a bounded operator with some applications |
| url | http://dx.doi.org/10.1155/2009/239025 |
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