A generalization of the punctured neighborhood theorem

Lee, Woo Young

doi:10.1090/S0002-9939-1993-1113645-7

A generalization of the punctured neighborhood theorem
HTML articles powered by AMS MathViewer

by Woo Young Lee

Proc. Amer. Math. Soc. 117 (1993), 107-109

DOI: https://doi.org/10.1090/S0002-9939-1993-1113645-7

PDF | Request permission

Abstract:

If $T \in \mathcal {L}(X)$ is regular on a Banach space $X$, with finite- dimensional intersection ${T^{ - 1}}(0) \cap T(X)$, and if $S,\;S’$ are invertible, commute with $T$ and have sufficiently small norm, then $\dim {(T - S’)^{ - 1}}(0) = \dim {(T - S)^{ - 1}}(0)$ and $\dim X/(T - S’)X = \dim X/(T - S)X$.

References

Seymour Goldberg, Unbounded linear operators: Theory and applications, McGraw-Hill Book Co., New York-Toronto-London, 1966. MR 0200692
Robin Harte, Invertibility and singularity for bounded linear operators, Monographs and Textbooks in Pure and Applied Mathematics, vol. 109, Marcel Dekker, Inc., New York, 1988. MR 920812
Harro G. Heuser, Functional analysis, A Wiley-Interscience Publication, John Wiley & Sons, Ltd., Chichester, 1982. Translated from the German by John Horváth. MR 640429
Woo Young Lee, Relatively open mappings, Proc. Amer. Math. Soc. 108 (1990), no. 1, 93–94. MR 984804, DOI 10.1090/S0002-9939-1990-0984804-6
Woo Young Lee, Boundaries of the spectra in ${\scr L}(X)$, Proc. Amer. Math. Soc. 116 (1992), no. 1, 185–189. MR 1094504, DOI 10.1090/S0002-9939-1992-1094504-4
C. R. Putnam, The spectra of operators having resolvents of first-order growth, Trans. Amer. Math. Soc. 133 (1968), 505–510. MR 229073, DOI 10.1090/S0002-9947-1968-0229073-2