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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


Analyticity of determinants of operators on a Banach space

Author: James S. Howland
Journal: Proc. Amer. Math. Soc. 28 (1971), 177-180
MSC: Primary 47A55; Secondary 47B10
MathSciNet review: 0417827
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Abstract: If $ F(z)$ is an analytic family of operators on a Banach space which is of finite rank for each $ z$, then rank $ F(z)$ is constant except for isolated points, and det $ (I + F(z))$ and tr $ F(z)$ are analytic. Similarly if $ F(z)$ is meromorphic.

References [Enhancements On Off] (What's this?)

  • [1] Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473 (34 #3324)
  • [2] A. F. Ruston, On the Fredholm theory of integral equations for operators belonging to the trace class of a general Banach space, Proc. London Math. Soc. (2) 53 (1951), 109–124. MR 0042612 (13,138c)
  • [3] A. F. Ruston, Auerbach’s theorem and tensor products of Banach spaces, Proc. Cambridge Philos. Soc. 58 (1962), 476–480. MR 0165346 (29 #2630)

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Additional Information

PII: S 0002-9939(1971)0417827-4
Keywords: Trace class of Banach space, operator-valued analytic functions, determinants, finite rank operators
Article copyright: © Copyright 1971 American Mathematical Society