On compact operators in the weak closure of the range of a derivation
Author:
Hong W. Kim
Journal:
Proc. Amer. Math. Soc. 40 (1973), 482-486
MSC:
Primary 47B47; Secondary 46L10
DOI:
https://doi.org/10.1090/S0002-9939-1973-0318956-8
MathSciNet review:
0318956
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Abstract | References | Similar Articles | Additional Information
Abstract: It is shown that if 
is a compact operator which commutes with a bounded operator 
on a Hilbert space 
and if 
is contained in the weak closure of the range of the derivation induced by 
, then 
is quasinilpotent.
- [1] J. H. Anderson, Derivation ranges and the identity, Bull. Amer. Math. Soc. 79 (1973), 705–708. MR 322518, https://doi.org/10.1090/S0002-9904-1973-13271-9
- [2] I. C. Gohberg and M. G. Kreĭn, Introduction to the theory of linear nonselfadjoint operators, Translated from the Russian by A. Feinstein. Translations of Mathematical Monographs, Vol. 18, American Mathematical Society, Providence, R.I., 1969. MR 0246142
- [3] Paul R. Halmos, A Hilbert space problem book, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR 0208368
- [4] Maurice Heins, Complex function theory, Pure and Applied Mathematics, Vol. 28, Academic Press, New York-London, 1968. MR 0239054
- [5] David C. Kleinecke, On operator commutators, Proc. Amer. Math. Soc. 8 (1957), 535–536. MR 87914, https://doi.org/10.1090/S0002-9939-1957-0087914-4
- [6] C. Pearcy, Some unsolved problems on operator theory, Studies in Operator Theory, Math. Assoc. Amer. (to appear).
- [7] Frédéric Riesz and Béla Sz.-Nagy, Leçons d’analyse fonctionnelle, Akadémiai Kiadó, Budapest, 1953 (French). 2ème éd. MR 0056821
- [8] F. V. Širokov, Proof of a conjecture of Kaplansky, Uspehi Mat. Nauk (N.S.) 11 (1956), no. 4(70), 167–168 (Russian). MR 0087913
- [9]
J. G. Stampfli, Derivations on
: The range (to appear).
- [10] J. P. Williams, On the range of a derivation, Pacific J. Math. 38 (1971), 273–279. MR 308809
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Additional Information
DOI:
https://doi.org/10.1090/S0002-9939-1973-0318956-8
Keywords:
Range of a derivation,
eigenvalues,
quasinilpotent operators,
norm closure,
weak operator topology,
commutant of an operator,
compact operator
Article copyright:
© Copyright 1973
American Mathematical Society
