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Proceedings of the American Mathematical Society

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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
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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., $ \vert\vert{K_n}\vert\vert \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.

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