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Monotone continuity of the spectral resolution and the eigenvalues

Published online by Cambridge University Press:  14 November 2011

Joachim Weidmann
Joachim Weidmann
Affiliation:
Fachbereich Mathematik der Universität, Robert-Mayer-Straße 6–10, D6000 Frankfurt am Main, B.R.D.
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Synopsis

Let (Tn) be a non-decreasing sequence of self-adjoint operators which are semi-bounded from below and converge to some self-adjoint operator T in the sense of strong resolvent convergence. For every λ which is eventually below the essential spectrum of Tn it is shown that ‖En(λ)−E(λ)‖→0 for n→∞, where En(·) and E(·) are the spectral resolution of Tn and T, respectively.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 85 , Issue 1-2 , 1980 , pp. 131 - 136
Copyright
Copyright © Royal Society of Edinburgh 1980

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References

1Fans, W. G.. Self-adjoint operators. Lecture Notes in Mathematics 433 (Berlin: Springer, 1976).Google Scholar
2Kato, T.. Perturbation Theory for Linear Operators, 2nd edn (Berlin: Springer, 1976).Google Scholar
3Weidmann, J.. Lineare Operatoren in Hilberträumen (Stuttgart: Teubner, 1976).Google Scholar