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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A formula for the resolvent of $ (-\Delta)\sp m+M\sp {2m}\sb q$ with applications to trace class


Author: Peter Takáč
Journal: Trans. Amer. Math. Soc. 303 (1987), 325-344
MSC: Primary 47F05; Secondary 35J05, 35P05, 47B10
MathSciNet review: 896025
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Abstract: We derive a formula for the resolvent of the elliptic operator $ H = {( - \Delta )^m} + M_q^{2m}$ on $ {L_2}({\mathbb{R}^N})$ in terms of bounded integral operators $ {S_\lambda }$ and $ {T_\lambda }$ whose kernels we know explicitly. We use this formula to specify the domain of the operator $ {A_\lambda } = (H + \lambda I){M_p}$ on $ {L_2}({\mathbb{R}^N})$, and to estimate the Hilbert-Schmidt norm of its inverse $ A_\lambda ^{ - 1}$, for $ \lambda \geqslant 0$. Finally we exploit the last two results to prove a trace class criterion for an integral operator $ K$ on $ {L_2}({\mathbb{R}^N})$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1987-0896025-0
PII: S 0002-9947(1987)0896025-0
Article copyright: © Copyright 1987 American Mathematical Society