Complete continuity of the inverse of a positive symmetric operator.
Author:
James P. Fink
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
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Abstract | References | Similar Articles | Additional Information
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 $H$.
- Kurt Friedrichs, Spektraltheorie halbbeschränkter Operatoren und Anwendung auf die Spektralzerlegung von Differentialoperatoren, Math. Ann. 109 (1934), no. 1, 465–487 (German). MR 1512905, DOI https://doi.org/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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Keywords:
Linear transformations on Hilbert space,
symmetric linear transformation,
positive linear transformation,
completely continuous linear transformation,
inverse transformation,
eigenvalues of completely continuous transformations,
compact linear transformation
Article copyright:
© Copyright 1970
American Mathematical Society