The distance from the Apostol spectrum

Kordula, V.; Müller, V.

doi:10.1090/S0002-9939-96-03306-0

The distance from the Apostol spectrum
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by V. Kordula and V. Müller PDF

Proc. Amer. Math. Soc. 124 (1996), 3055-3061 Request permission

Abstract:

If $T$ is an s-regular operator in a Banach space (i.e. $T$ has closed range and $N(T)\subset R^{\infty }(T)$) and $\gamma (T)$ is the Kato reduced minimum modulus, then \begin{equation*}\lim _{n\to \infty }\gamma (T^{n})^{1/n}=\sup \{r: T-\lambda { \operatorname {is s-regular for }}|\lambda |<r\}. \end{equation*}

References

Constantin Apostol, The reduced minimum modulus, Michigan Math. J. 32 (1985), no. 3, 279–294. MR 803833, DOI 10.1307/mmj/1029003239
K.-H. Förster and M. A. Kaashoek, The asymptotic behaviour of the reduced minimum modulus of a Fredholm operator, Proc. Amer. Math. Soc. 49 (1975), 123–131. MR 372660, DOI 10.1090/S0002-9939-1975-0372660-0
M. A. Gol′dman and S. N. Kračkovskiĭ, The stability of certain properties of a closed linear operator, Dokl. Akad. Nauk SSSR 209 (1973), 769–772 (Russian). MR 0322548
Sandy Grabiner, Uniform ascent and descent of bounded operators, J. Math. Soc. Japan 34 (1982), no. 2, 317–337. MR 651274, DOI 10.2969/jmsj/03420317
M. A. Kaashoek, Stability theorems for closed linear operators, Nederl. Akad. Wetensch. Proc. Ser. A 68=Indag. Math. 27 (1965), 452–466. MR 0181903, DOI 10.1016/S1385-7258(65)50048-2
Tosio Kato, Perturbation theory for nullity, deficiency and other quantities of linear operators, J. Analyse Math. 6 (1958), 261–322. MR 107819, DOI 10.1007/BF02790238
Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473
J.P. Labrouse, Les opérateurs quasi-Fredholm: une généralisation des opérateurs semi- Fredholm, Rend. Circ. Math. Palermo 29 (1989), 69–105.
Endre Makai Jr. and Jaroslav Zemánek, The surjectivity radius, packing numbers and boundedness below of linear operators, Integral Equations Operator Theory 6 (1983), no. 3, 372–384. MR 701026, DOI 10.1007/BF01691904
Mostafa Mbekhta, Résolvant généralisé et théorie spectrale, J. Operator Theory 21 (1989), no. 1, 69–105 (French). MR 1002122
M. Mbekhta and A. Ouahab, Opérateur s-régulier dans un espace de Banach et théorie spectrale, vol. 22, No. XII, Pub. Irma, Lille, 1990.
Ouahab Abdelmalek and Mbekhta Mostafa, Contribution à la théorie spectrale généralisée dans les espaces de Banach, C. R. Acad. Sci. Paris Sér. I Math. 313 (1991), no. 12, 833–836 (French, with English summary). MR 1138559
Vladimír Müller, The inverse spectral radius formula and removability of spectrum, Časopis Pěst. Mat. 108 (1983), no. 4, 412–415 (English, with Russian summary). MR 727539, DOI 10.21136/CPM.1983.118186
—, On the regular spectrum, J. Operator Theory (to appear).
Vladimir Rakočević, Generalized spectrum and commuting compact perturbations, Proc. Edinburgh Math. Soc. (2) 36 (1993), no. 2, 197–209. MR 1221044, DOI 10.1017/S0013091500018332
Pierre Saphar, Contribution à l’étude des applications linéaires dans un espace de Banach, Bull. Soc. Math. France 92 (1964), 363–384 (French). MR 187095, DOI 10.24033/bsmf.1612
Jaroslav Zemánek, The stability radius of a semi-Fredholm operator, Integral Equations Operator Theory 8 (1985), no. 1, 137–144. MR 775217, DOI 10.1007/BF01199985