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
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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
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