Hostname: page-component-77c89778f8-7drxs Total loading time: 0 Render date: 2024-07-16T14:49:21.277Z Has data issue: false hasContentIssue false
  • English
  • Français

Regularizers of Closed Operators

Published online by Cambridge University Press:  20 November 2018

C.-S. Lin
C.-S. Lin*
Affiliation:
University of New Brunswick, Fredericton, N.B.
Rights & Permissions [Opens in a new window]

Extract

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.

Let X and Y be two Banach spaces and let B(X, Y) denote the set of bounded linear operators with domain X and range in 7. For T∈B(X, Y), let N(T) denote the null space and R(T) the range of T. J. I. Nieto [5, p. 64] has proved the following two interesting results. An operator T∈B(X, Y) has a left regularizer, i.e., there exists a Q∈B(Y, X) such that QT=I+A, where I is the identity on X and A∈B(X, X) is a compact operator, if and only if dim N(T)<∞ and R(T) has a closed complement.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 17 , Issue 1 , 01 March 1974 , pp. 67 - 71
Copyright
Copyright © Canadian Mathematical Society 1974

References

1. Dieudonné, J., Foundations of modern analysis, Academic Press, New York (1960).Google Scholar
2. Goldberg, S., Unbounded linear operators, McGraw-Hill, New York (1966).Google Scholar
3. Kato, T., Perturbation theory for nullity, deficiency and other quantities of linear operators, J. Analyse Math. 6, 261-322 (1958).Google Scholar
4. Kato, T., Perturbation theory for linear operators, Berlin-Heidelberg-New York, Springer- Verlag, New York (1966).Google Scholar
5. Keito, J. I., On Fredholm operators and the essential spectrum of singular integral operators, Math. Ann. 178, 62-77 (1968).Google Scholar
6. Yood, B., Properties of linear transformations preserved under addition of a completely continuous transformation, Duke Math. J. 18, 599-612 (1951).Google Scholar