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

Goldberg, Seymour

doi:10.1090/S0002-9939-1974-0346579-4

Perturbations of semi-Fredholm operators by operators converging to zero compactly
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by Seymour Goldberg PDF

Proc. Amer. Math. Soc. 45 (1974), 93-98 Request permission

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.

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