Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-26T14:30:52.478Z Has data issue: false hasContentIssue false

Doubling perturbation sizes and preservation of operator indices in normed linear spaces

Published online by Cambridge University Press:  24 October 2008

Karl Gustafson
Affiliation:
University of Minnesota

Abstract

We extend the index theory of Fredholm operators in Banach spaces to general operators in normed linear spaces. In so doing we make use of a doubling technique, which we also apply to obtain improved results for the perturbation of semigroups and sesquilinear forms.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1969

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Gustafson, K.A perturbation lemma. Bull. Amer. Math. Soc. 72 (1966), 334338.CrossRefGoogle Scholar
(2)Kato, T.Perturbation theory for nullity, deficiency, and other quantities of linear operators. J. Analyse Math. 6 (1958), 261322.CrossRefGoogle Scholar
(3)Kato, T.Perturbation theory for linear operators (Springer-Verlag; Berlin, 1966).Google Scholar
(4)Goldberg, S.Unbounded linear operators (McGraw-Hill; New York, 1966).Google Scholar
(5)Krein, M. G., Krasnosel'skii, M. A. and Milman, D. C.On the defect numbers of linear operators in Banach space and on some geometric problems. Sb. Tr. Inst. Mat. Akad. Nauk, Ukr. SSR 11 (1948), 97112.Google Scholar
(6)Sz.-Nagy, B.Perturbations des transformations linéaire fermées. Acta Sci. Math. Szeged 14 (1951), 125137.Google Scholar
(7)Nelson, E.Feynman integrals and the Schrödinger equation. J. Mathematical Phys. 5 (1964), 332343.CrossRefGoogle Scholar
(8)Lumer, G. and Phillips, R. S.Dissipative operators in a Banach space. Pacific J. Math. 11 (1961), 679698.CrossRefGoogle Scholar
(8a)Sato, K.On the generators of non-negative contraction semigroups in Banach lattices. J. Math. Soc. Japan 20 (1968), 423436.CrossRefGoogle Scholar
(9)Webb, J. H.Perturbation theory for a linear operator. Proc. Cambridge Philos. Soc. 63 (1967), 1120.CrossRefGoogle Scholar
(10)Goldberg, S.Ranges and inverses of perturbed linear operators. Pacific J. Math. 9 (1959), 701706.CrossRefGoogle Scholar
(11)Kaashoek, M. A.Closed linear operators in Banach spaces. Proc. Kon. Nederl. Akad. Wetensch. Ser. A 68 (1965), 405414.CrossRefGoogle Scholar
(12)Yosida, K.A perturbation theorem for semi-groups of linear operators. Proc. Japan Acad. 41 (1965), 645647.Google Scholar
(13)Dorroh, J. R.Contraction semigroups in a function space. Pacific J. Math. 19 (1966), 3538.CrossRefGoogle Scholar
(14)Gustafson, K.A note on left multiplication of semigroup generators. Pacific J. Math. 24 (1968), 463465.CrossRefGoogle Scholar
(15)Gustafson, K.Positive (noncommuting) operator products and semigroups. Math. Z. 105 (1968), 160172.CrossRefGoogle Scholar
(16)Gustafson, K. and Lumer, G. Multiplicative perturbation of semigroups. To appear.Google Scholar
(17)Moore, R. T.Duality methods and perturbation of semigroups. Bull. Amer. Math. Soc. 73 (1967), 548553.CrossRefGoogle Scholar
(18)Oliver, R. Contributions to perturbation theory for linear operators with closed range. Thesis, Univ. of Maryland (1966).Google Scholar
(19)Gustafson, K.State diagrams for Hilbert space operators. J. Math. Mech. 18 (1968), 3346.Google Scholar
(20)Beals, R. W.A note on the adjoint of a perturbed operator. Bull. Amer. Math. Soc. 70 (1964), 314315.CrossRefGoogle Scholar
(21)Gustafson, K.The angle of an operator and positive operator products, Bull. Amer. Math. Soc. 74, (1968), 488492.CrossRefGoogle Scholar