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Algebra i Analiz, 2024, Volume 36, Issue 1, Pages 95–161 (Mi aa1902)  

Research Papers

Threshold approximations for functions of a factorized operator family

M. A. Dorodnyi, T. A. Suslina

Saint Petersburg State University
References:
Abstract: In a Hilbert space $\mathfrak H$, we consider a family of selfadjoint operators (a quadratic operator pencil) $A(t)$, $t\in \mathbb{R}$, of the form $A(t) = X(t)^* X(t)$, where $X(t) = X_0 + t X_1$. It is assumed that the point $\lambda_0=0$ is an isolated eigenvalue of finite multiplicity for the operator $A(0)$. Let $F(t)$ be the spectral projection of the operator $A(t)$ for the interval $[0,\delta]$. Using approximations for $F(t)$ and $A(t)F(t)$ for $|t| \leqslant t_0$ (the so-called threshold approximations), we obtain approximations in the operator norm on $\mathfrak H$ for the operators $\cos ( \tau A(t)^{1/2})$ and $A(t)^{-1/2}\sin ( \tau A(t)^{1/2})$, $\tau \in \mathbb{R}$. The numbers $\delta$ and $t_0$ are controlled explicitly. Next, we study the behavior for small $\varepsilon >0$ of the operators $\cos (\varepsilon^{-1} \tau A(t)^{1/2})$ and $A(t)^{-1/2}\sin (\varepsilon^{-1}\tau A(t)^{1/2})$ multiplied by the “smoothing factor” $\varepsilon^q (t^2 + \varepsilon^2)^{-q/2}$ with a suitable $q>0$. The obtained approximations are given in terms of the spectral characteristics of the operator $A(t)$ near the lower edge of the spectrum. The results are aimed at application to homogenization of hyperbolic equations with periodic rapidly oscillating coefficients.
Keywords: homogenization, quadratic operator pencils, threshold approximations, analytic perturbation theory.
Funding agency Grant number
Russian Science Foundation 22-11-00092
Received: 06.11.2023
Document Type: Article
Language: Russian
Citation: M. A. Dorodnyi, T. A. Suslina, “Threshold approximations for functions of a factorized operator family”, Algebra i Analiz, 36:1 (2024), 95–161
Citation in format AMSBIB
\Bibitem{DorSus24}
\by M.~A.~Dorodnyi, T.~A.~Suslina
\paper Threshold approximations for functions of a factorized operator family
\jour Algebra i Analiz
\yr 2024
\vol 36
\issue 1
\pages 95--161
\mathnet{http://mi.mathnet.ru/aa1902}
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