The spectral diameter in Banach algebras

Grabiner, Sandy

doi:10.1090/S0002-9939-1984-0735564-2

The spectral diameter in Banach algebras

Author: Sandy Grabiner
Journal: Proc. Amer. Math. Soc. 91 (1984), 59-63
MSC: Primary 46H05; Secondary 47A65
DOI: https://doi.org/10.1090/S0002-9939-1984-0735564-2
MathSciNet review: 735564
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The element is in the center of the Banach algebra modulo its radical if and only if there is an upper bound for the diameters of the spectra of ${a^\_}ua{u^{ - 1}}$ for invertible. Applications of this result are given to general Banach algebras and to the essential spectrum of operators on a Hilbert Space.

[1] B. Aupetit, Propriétés spectrales des algebres de Banach, Lecture Notes in Math., vol. 735, Springer-Verlag, Berlin and New York, 1979. MR 549769 (81i:46055)
[2] B. Aupetit and J. Zemánek, Local behaviour of the spectral radius in Banach algebras, J. London Math. Soc. 23 (1981), 171-178. MR 602249 (83d:46056)
[3] F. F. Bonsall and J. Duncan, Complete normed algebras, Springer-Verlag, New York, 1973. MR 0423029 (54:11013)
[4] A. Brown, C. Pearcy and Norberto Salinas, Perturbations by nilpotent operators on Hilbert space, Proc Amer. Math. Soc. 41 (1973), 530-540. MR 0374955 (51:11151)
[5] S. R. Caradus, W. E. Pfaffenberger and B. Yood, Calkin algebras and algebras of operators on Banach spaces, Lecture Notes in Pure and Appl. Math., vol. 9, Dekker, New York, 1974. MR 0415345 (54:3434)
[6] C. Le Page, Sur quelques conditions entraînant la commutativité dans les algebres de Banach, C. R. Acad. Sci. Paris Sér. A-B 265 (1967), 235-237. MR 0226409 (37:1999)
[7] J. D. Newburgh, The variation of spectra, Duke Math. J. 18 (1951), 165-176. MR 0051441 (14:481b)
[8] A. M. Sinclair, Automatic continuity of linear operators, London Math. Soc. Lecture Note Series, No. 21, Cambridge Univ. Press, Cambridge, New York, Melbourne, 1976. MR 0487371 (58:7011)
[9] T. T. West, The decomposition of Riesz operators, Proc. London Math. Soc. (3) 16 (1966), 737-752. MR 0198258 (33:6417)
[10] J. Zemánek, Properties of the spectral radius in Banach algebras, Inst. of Math., Polish Acad. Sci. Preprint 136, 1978; Proc. Spectral Theory Semester, Banach Center Publ., vol. 8, PWN, Warsaw, 1982. MR 738318 (85d:46064)