Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-25T08:00:06.371Z Has data issue: false hasContentIssue false

The generalized condition numbers of bounded linear operators in Banach spaces

Published online by Cambridge University Press:  09 April 2009

Guoliang Chen
Affiliation:
Department of Mathematics, East China Normal University, Shanghai, 200062 P.R., China
Yimin Wei
Affiliation:
Department of Mathematics, Fudan UniversityShanghai 200433 P. R., China
Yifeng Xue
Affiliation:
Department of Mathematics, East China University of Science and Technology, Shanghai 200237 P.R., China e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

For any bounded linear operator A in a Banach space, two generalized condition numbers, k(A) and k(A) are defined in this paper. These condition numbers may be applied to the perturbation analysis for the solution of ill-posed differential equations and bounded linear operator equations in infinite dimensional Banach spaces. Different expressions for the two generalized condition numbers are discussed in this paper and applied to the perturbation analysis of the operator equation.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 2004

References

[1]Chen, G., ‘Minimum property in some problems with condition number of square matrices’, Comm. Appl. Math. Comput. 1 (1988), 7480 (Chinese).Google Scholar
[2]Chen, G. and Chen, D., ‘On the minimum property of condition number for doubly perturbed linear equations’, J. Math. Applicata 3 (1989), 6976 (Chinese).Google Scholar
[3]Chen, G. and Xue, Y., ‘Perturbation analysis for the operator equation Tx = b in Banach spaces’, J. Math. Anal. Appl. 212 (1997), 107125.CrossRefGoogle Scholar
[4]Demko, S., ‘Condition numbers of rectangular systems and bounds for generalized inverses’, Linear Algebra Appl. 78 (1986), 199206.CrossRefGoogle Scholar
[5]Dor, L. E., ‘On projections in L1’, Ann. of Math. (2) 102 (1975), 463474.CrossRefGoogle Scholar
[6]Dunford, N. and Schwartz, J. T., Linear operators. Part 3: Spectral operators (Wiley, New York, 1971).Google Scholar
[7]Kato, T., Perturbation theory for linear operators, 2nd edition (Springer, Berlin, 1984).Google Scholar
[8]Kuang, J., ‘The ω-condition number of linear operators’, Numer Math. J. Chinese Univ. 1 (1980), 1118 (Chinese).Google Scholar
[9]Pragua, J. R., ‘New condition number for matrices and linear systems’, Computing 41 (1989), 211213.Google Scholar
[10]Taylor, A. E., Introduction to functional analysis (Wiley, New York, 1958).Google Scholar
[11]Wolkowicz, H. and Zlobec, S., ‘Calculating the best approximate solution of an operator equation’, Math. Comp. 32 (1978), 11831213.CrossRefGoogle Scholar
[12] Xue, Y., ‘The reduced minimum modulus of elements in C*-algebras’, preprint.Google Scholar