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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Complete continuity of the inverse of a positive symmetric operator.
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by James P. Fink
Proc. Amer. Math. Soc. 25 (1970), 147-150
DOI: https://doi.org/10.1090/S0002-9939-1970-0257797-4

Abstract:

Let $A$ be a symmetric positive definite linear transformation defined on a dense subset of a Hilbert space $H$, and let ${H_A}$. be the Hilbert space completion of the domain of $A$ with respect to the inner product ${(u,v)_A} = (Au,v)$. It is shown that the inverse of $A$ is completely continuous on ${H_A}$ if and only if it is completely continuous on $H$.
References
  • Kurt Friedrichs, Spektraltheorie halbbeschränkter Operatoren und Anwendung auf die Spektralzerlegung von Differentialoperatoren, Math. Ann. 109 (1934), no. 1, 465–487 (German). MR 1512905, DOI 10.1007/BF01449150
  • Frigyes Riesz and Béla Sz.-Nagy, Functional analysis, Frederick Ungar Publishing Co., New York, 1955. Translated by Leo F. Boron. MR 0071727
  • S. G. Mikhlin, The problem of the minimum of a quadratic functional, Holden-Day Series in Mathematical Physics, Holden-Day, Inc., San Francisco, Calif.-London-Amsterdam, 1965. Translated by A. Feinstein. MR 0171196
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Bibliographic Information
  • © Copyright 1970 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 25 (1970), 147-150
  • MSC: Primary 47.45
  • DOI: https://doi.org/10.1090/S0002-9939-1970-0257797-4
  • MathSciNet review: 0257797