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St. Petersburg Mathematical Journal

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Cwikel type estimate as a consequence of certain properties of the heat kernel


Author: V. A. Sloushch
Translated by: A. Kiselev
Original publication: Algebra i Analiz, tom 25 (2013), nomer 5.
Journal: St. Petersburg Math. J. 25 (2014), 835-854
MSC (2010): Primary 35K08; Secondary 47B10
DOI: https://doi.org/10.1090/S1061-0022-2014-01318-0
Published electronically: July 18, 2014
MathSciNet review: 3184610
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Abstract: Estimates for the singular values of the operator $ \mathbb{T}_{fg}:=f(H)g(x)$ are investigated for suitable functions $ f(\lambda )$, $ \lambda \in \mathbb{R}$, $ g(x)$, $ x\in \mathbb{R}^{d}$, and a selfadjoint operator $ H$ in $ L_{2}(\mathbb{R}^{d})$. It is assumed that the kernel of the semigroup $ e^{-tH}$ satisfies special conditions. Power-like estimates for the singular values of the operator $ \mathbb{T}_{fg}$ are obtained, in particular, in the case where $ \mathbb{T}_{fg}\in \mathfrak{S}_{2}$. Conditions for the operator $ \mathbb{T}_{fg}$ to belong to the trace class are established. Neither any smoothness conditions for the kernel of the operator $ \mathbb{T}_{fg}$, nor any knowledge of the (partial) diagonalization of the operator $ H$ are required. The results admit further refinement under additional conditions imposed on the generalized eigenfunctions of the operator $ H$.


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Additional Information

V. A. Sloushch
Affiliation: Department of Physics, St. Petersburg State University, Ul′yanovskaya 2, Staryi Peterhof, St. Petersburg 198904, Russia
Email: vsloushch@list.ru

DOI: https://doi.org/10.1090/S1061-0022-2014-01318-0
Keywords: Integral operators, estimates for the singular values, classes of compact operators
Received by editor(s): September 21, 2012
Published electronically: July 18, 2014
Additional Notes: Supported by RFBR (grant no. 11-01-00458-a) and by the Leading Scientific Schools Support Programme (grant no. NSh-357.2012.1).
Article copyright: © Copyright 2014 American Mathematical Society

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