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Asymptotically quasi-compact products of bounded linear operators

Published online by Cambridge University Press:  09 April 2009

Anthony F. Ruston
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
Information
Journal of the Australian Mathematical Society , Volume 16 , Issue 3 , November 1973 , pp. 286 - 289
Copyright
Copyright © Australian Mathematical Society 1973

References

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