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A comparison theorem for selfadjoint operators

Author: Amin Boumenir
Journal: Proc. Amer. Math. Soc. 111 (1991), 161-175
MSC: Primary 47B25; Secondary 34L40, 47A70, 47E05
MathSciNet review: 1021896
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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

$\displaystyle (d{\Gamma _1}/d{\Gamma _2})({L_2}) = \overline V V',$

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

$\displaystyle y = V\varphi ,$

where $ {L_2}y = \lambda y$ and $ {L_1}\varphi = \lambda \varphi $.

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

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Article copyright: © Copyright 1991 American Mathematical Society