Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Perturbations of semi-Fredholm operators by operators converging to zero compactly

Author: Seymour Goldberg
Journal: Proc. Amer. Math. Soc. 45 (1974), 93-98
MSC: Primary 47B30
MathSciNet review: 0346579
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $\{ {K_n}\}$ be a sequence of bounded linear operators mapping a Banach space $X$ into a Banach space such that ${K_n} \to 0$ strongly and $\{ {K_n}{x_n}\}$ is relatively compact for every bounded sequence $\{ {x_n}\} \subset X$; e.g., $||{K_n}|| \to 0$. Given $T$ a semi-Fredholm operator, it is shown that for all sufficiently large $n,T + {K_n}$ has nullity and deficiency not exceeding that of $T$ while the index of $T + {K_n}$ equals that of $T$. Properties of the minimum modulus of $T + {K_n}$ are also given.

References [Enhancements On Off] (What's this?)

  • Philip M. Anselone, Collectively compact operator approximation theory and applications to integral equations, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1971. With an appendix by Joel Davis; Prentice-Hall Series in Automatic Computation. MR 0443383
  • I. C. Gohberg and M. G. Kreĭn, The basic propositions on defect numbers, root numbers and indices of linear operators, Amer. Math. Soc. Transl. (2) 13 (1960), 185–264. MR 0113146
  • Seymour Goldberg, Unbounded linear operators: Theory and applications, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0200692
  • Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473
  • L. S. Rakovščik, Stability of the index and semi-stability of the defect numbers under compact approximation, Sibirsk. Mat. Ž. 13 (1972), 630–637 (Russian). MR 0350471

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 47B30

Retrieve articles in all journals with MSC: 47B30

Additional Information

Keywords: Semi-Fredholm, index, nullity, deficiency, minimum modulus
Article copyright: © Copyright 1974 American Mathematical Society