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
  • [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
  • [3] A. F. Ruston, Auerbach’s theorem and tensor products of Banach spaces, Proc. Cambridge Philos. Soc. 58 (1962), 476–480. MR 0165346

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1971-0417827-4
Keywords: Trace class of Banach space, operator-valued analytic functions, determinants, finite rank operators
Article copyright: © Copyright 1971 American Mathematical Society