Geodesic
Criteria for stability of the point spectrum under completely continuous perturbations
Matematičeskie zametki, Tome 18 (1975) no. 4, pp. 601-607.

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We show that a number $\lambda$ is an eigenvalue of the operator $T+C$ for an arbitrary compact perturbation $C$ if and only if the operator $T-\lambda I$ is semi-Fredholm and $\mathrm{ind}\,(T-\lambda I)>0$. 
@article{MZM_1975_18_4_a13,
     author = {L. N. Nikol'skaya},
     title = {Criteria for stability of the point spectrum under completely continuous perturbations},
     journal = {Matemati\v{c}eskie zametki},
     pages = {601--607},
     publisher = {mathdoc},
     volume = {18},
     number = {4},
     year = {1975},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1975_18_4_a13/}
}
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L. N. Nikol'skaya. Criteria for stability of the point spectrum under completely continuous perturbations. Matematičeskie zametki, Tome 18 (1975) no. 4, pp. 601-607. http://geodesic.mathdoc.fr/item/MZM_1975_18_4_a13/