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

  • [1] K. Friedrichs, Spektraltheorie halbbeschränkter Operatoren und Anwendung auf die Spektralzerlegung von Differentialoperatoren, Math. Ann. 109 (1934), 465. MR 1512905
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  • [3] S. G. Mihlin, The problem of the minimum of a quadratic functional, GITTL, Moscow, 1952; English transl., Holden-Day Series in Mathematical Physics, Holden-Day, San Francisco, Calif., 1965. MR 16, 41; MR 30 #1427. MR 0171196 (30:1427)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1970-0257797-4
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

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