Almost-everywhere discontinuity of the spectral radius

Ransford, Thomas

doi:10.1090/S0002-9939-00-05617-3

Almost-everywhere discontinuity of the spectral radius
HTML articles powered by AMS MathViewer

by Thomas J. Ransford

Proc. Amer. Math. Soc. 129 (2001), 749-751

DOI: https://doi.org/10.1090/S0002-9939-00-05617-3

Published electronically: September 19, 2000

PDF | Request permission

Abstract:

Let $\rho (T)$ denote the spectral radius of an operator $T$. We construct operators $S$ and $T$ on $\ell ^2$ such that $\lambda \mapsto \rho (T-\lambda S)$ is discontinuous almost everywhere on the unit disk.

References

Bernard Aupetit, Propriétés spectrales des algèbres de Banach, Lecture Notes in Mathematics, vol. 735, Springer, Berlin, 1979 (French). MR 549769
W. T. Gowers and B. Maurey, The unconditional basic sequence problem, J. Amer. Math. Soc. 6 (1993), no. 4, 851–874. MR 1201238, DOI 10.1090/S0894-0347-1993-1201238-0
W. K. Hayman and P. B. Kennedy, Subharmonic functions. Vol. I, London Mathematical Society Monographs, No. 9, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1976. MR 0460672
K. Kuratowski, Topology. Vol. II, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw, 1968. New edition, revised and augmented; Translated from the French by A. Kirkor. MR 0259835
Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. I, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 92, Springer-Verlag, Berlin-New York, 1977. Sequence spaces. MR 0500056
Vladimír Müller, On discontinuity of the spectral radius in Banach algebras, Comment. Math. Univ. Carolinae 18 (1977), no. 3, 591–598. MR 500157
Charles E. Rickart, General theory of Banach algebras, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0115101