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Another note on Weyl’s theorem. (English) Zbl 0873.47001
Summary: “Weyl’s theorem holds” for an operator \(T\) on a Banach space \(X\) when the complement in the spectrum of the “Weyl spectrum” coincides with the isolated points of spectrum which are eigenvalues of finite multiplicity. This is close to, but not quite the same as, equality between the Weyl spectrum and the “Browder spectrum”, which in turn ought to, but does not, guarantee the spectral mapping theorem for the Weyl spectrum of polynomials in \(T\). In this note we try to explore these distinctions.

47A10 Spectrum, resolvent
47A60 Functional calculus for linear operators
Full Text: DOI
[1] S. K. Berberian, An extension of Weyl’s theorem to a class of not necessarily normal operators, Michigan Math. J. 16 (1969), 273 – 279. · Zbl 0175.13603
[2] S. K. Berberian, The Weyl spectrum of an operator, Indiana Univ. Math. J. 20 (1970/1971), 529 – 544. · Zbl 0203.13401 · doi:10.1512/iumj.1970.20.20044 · doi.org
[3] Pet\(^{\prime}\)r Bŭrnev, On the gravitational potential of homogeneous polygons, Bŭlgar. Akad. Nauk. Otd. Mat. Fiz. Nauk. Izv. Mat. Inst. 8 (1964), 13 – 22 (Bulgarian, with Russian and French summaries).
[4] Robin Harte, Fredholm, Weyl and Browder theory, Proc. Roy. Irish Acad. Sect. A 85 (1985), no. 2, 151 – 176. · Zbl 0567.47001
[5] Robin Harte, Regular boundary elements, Proc. Amer. Math. Soc. 99 (1987), no. 2, 328 – 330. · Zbl 0617.46052
[6] 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. · Zbl 0636.47001
[7] Woo Young Lee and Hong Youl Lee, On Weyl’s theorem, Math. Japon. 39 (1994), no. 3, 545 – 548. · Zbl 0814.47002
[8] W.Y. Lee and S.H. Lee, A spectral mapping theorem for the Weyl spectrum, Glasgow Math. Jour. 38 (1996), 61-64. CMP 1996:8
[9] Kirti K. Oberai, On the Weyl spectrum, Illinois J. Math. 18 (1974), 208 – 212. · Zbl 0277.47002
[10] Kirti K. Oberai, On the Weyl spectrum. II, Illinois J. Math. 21 (1977), no. 1, 84 – 90. · Zbl 0358.47004
[11] Carl M. Pearcy, Some recent developments in operator theory, American Mathematical Society, Providence, R.I., 1978. Regional Conference Series in Mathematics, No. 36. · Zbl 0444.47001
[12] J. G. Stampfli, Errata: Hyponormal operators and spectral density, Trans. Amer. Math. Soc. 115 (1965), 550. · Zbl 0139.31201
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