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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 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A comparison theorem for selfadjoint operators
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by Amin Boumenir PDF
Proc. Amer. Math. Soc. 111 (1991), 161-175 Request permission

Abstract:

In this work we shall establish a result concerning the spectral theory of differential operators. Let ${L_1}$ and ${L_2}$ be two self-adjoint operators acting in two different Hubert spaces. Then under some conditions we shall prove that \[ (d{\Gamma _1}/d{\Gamma _2})({L_2}) = \overline V V’,\] where ${\Gamma _1}(\lambda )$ and ${\Gamma _2}(\lambda )$ are the spectral functions associated with ${L_1}$ and ${L_2}$ respectively. $V$ is the shift operator mapping the set of generalized eigenfunctions of ${L_1}$ into the set of generalized eigenfunctions of ${L_2}$, that is \[ y = V\varphi ,\] where ${L_2}y = \lambda y$ and ${L_1}\varphi = \lambda \varphi$.
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Additional Information
  • © Copyright 1991 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 111 (1991), 161-175
  • MSC: Primary 47B25; Secondary 34L40, 47A70, 47E05
  • DOI: https://doi.org/10.1090/S0002-9939-1991-1021896-3
  • MathSciNet review: 1021896