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Asymptotically quasi-compact products of bounded linear operators

Published online by Cambridge University Press:  09 April 2009

Anthony F. Ruston
Affiliation:
Mathematics Insitute University of Warwick Coventry, CV4 7AL
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It is known (see, for instance, [1] p. 64, [6] p. 264) that, if A and B are bounded linear operators on a Banach space into itself (or, more generally, if A is a bounded linear operator on into a Banach space and B is a bounded linear operator on into), then AB and BA have the same spectrum except (possibly) for zero. In the present note, it is shown that AB is asymptotically quasi-compact if and only if BA is asymptotically quasi-compact, and that then any Fredholm determinant for AB is a Fredholm determinant for BA and vice versa.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1973

References

[1]Bonsall, F. F. and Duncan, J., Numerical Ranges of Operators on Normed Spaces and of Elements of Normed Algebras (London Math. Soc. Lecture Note Series No. 2, C. U. P., 1971). MR 44 # 5779.Google Scholar
[2]Dunford, Nelson and Schwartz, Jacob T., Linear Operators. Part I: General Theory (Interscience, New York, 1958). MR 22 # 8302.Google Scholar
[3]Ruston, A. F., ‘Formulae of Fredholm type for compact linear operators on a general Banach space’, Proc. London Math. Soc. (3) 3 (1953), 368377. MR 15, 134.CrossRefGoogle Scholar
[4]Ruston, A. F., ‘Operators with a Fredholm theory’, J. London Math. Soc. 29 (1954), 318326.CrossRefGoogle Scholar
[5]Ruston, A. F., ‘Fredholm formulae and the Riesz theory’, Compositio Math. 18 (1967), 2548. MR 36 # 4383.Google Scholar
[6]Taylor, Angus E., Introduction to Functional Analysis (Wiley, New York, 1958). MR 20 # 5411.Google Scholar