Abstract
For two bounded selfadjoint operators A and A + B with the same essential spectra, there exists a unitary operator U such that B − (UAU* − A) is compact (Weyl-Von Neumann). More generally, an operator B in B(H) is compact iff σe (A + B) = σe (A) for all A ∈ B(H) (Gustafson-Weidmann), and in fact one needs only σ(A + B) ∩ σ (A) not empty for all A ∈ B(H) (Dyer, Porcelli, Rosenfeld). Aiken (Is. Math. J., 1976) and Zemanek (Studia Math., to appear) have studied the question of when for an arbitrary Banach algebra with identity the last condition guarantees that B is in some proper two-sided ideal. We give new results for this question, including a number of examples.
Partially supported by Mashad University.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
K. Gustafson, Weyl’s Theorems, Proc. Oberwolfach Conf. on Linear Operators and Approximation, 1971, International Series of Numerical Mathematics, 20, Birkhauser-Verlag (1972), 80–93.
H. Weyl, Über beschränkte quadratische Formen, deren Differenz Vollsteting ist, Rend. cric. Math Palermo (1909)., 373–392.
J. Von Neumann, Charakterissierung des spektrums eienes integral operators, Actualites Sci. Indust. 229 (1935), 1–20.
K. Gustafson and J. Weidmann, On the essential spectrum, J. Math. Anal. Applic, 25 (1969), 121–127.
J. Zèmannek, Spectral characterization of two sided ideals in Banach algebras (to appear).
G. Aiken, Ph.D. dissertation, Louisiana State University, Baton Rouge, 1972.
G. Aiken, A problem of Dyer, Porcelli, and Rosenfeld, Israel J. Math., vol. 25, Nos. 3–4 (1976), 191–197.
J. Dyer, P. Porcelli, and M. Rosenfeld, Spectral characterization of two sided ideals in B(H), Israel J. Math. 10 (1971), 26–31.
C.E. Rickart, General Theory of Banach Algebras, D. Von Nostrand, Princeton (1960).
E.F. Bonsall and J. Duncan, Complete Normed Algebras, Springer-Verlag, Heidelberg 1973.
M. Naimark, Normed Rings, Nordhoff, Groningen, Netherlands (1959).
M. Rosenbloom, On the operator equation BX − XA = Q, Duke Math. J. 23, (1956), 263–270.
M. Seddighin, Ph.D. dissertation, University of Colorado at Boulder, to appear.
K. Gustafson, R.D. Goodrich, B. Misra, Irreversibility and Stochasticity of Chemical Processes, these proceedings.
I. Segal, The group ring of a locally compact group I, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 348–352.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1981 Plenum Press, New York
About this chapter
Cite this chapter
Seddighin, M., Gustafson, K. (1981). On a Generalized Weyl-Von Neumann Converse Theorem. In: Gustafson, K.E., Reinhardt, W.P. (eds) Quantum Mechanics in Mathematics, Chemistry, and Physics. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-3258-9_22
Download citation
DOI: https://doi.org/10.1007/978-1-4613-3258-9_22
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4613-3260-2
Online ISBN: 978-1-4613-3258-9
eBook Packages: Springer Book Archive