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 Mat. Sb., 2006, Volume 197, Number 7, Pages 29–46 (Mi msb1591)

A generalization of the concept of sectorial operator

M. F. Gorodnii, A. V. Chaikovskii

National Taras Shevchenko University of Kyiv

Abstract: Let $B$ be a Banach space and $G\colon[0,+\infty)\to(0,+\infty)$ a non-increasing function such that $G(t)\to0$ as $t\to\infty$ and $1/G$ is a Lipschitz function on $[0,+\infty)$.
A linear operator $T\colon D(T)\subset B\to B$ is said to be $G$-sectorial if there exist constants $a\in\mathbb R$ and $\varphi\in(0,\pi/2)$ such that the spectrum of $T$ lies in the set
$$S_{a,\varphi}:=ż\in\mathbb C\mid z\ne a, \lvert\arg(z-a)\rvert<\varphi\}$$
and
$$there exists M>0\quad such that \|R_\lambda(T)\|\le MG(|\lambda-a|) for \lambda\notin S_{a,\varphi},$$
where $R_\lambda(T)$ is the resolvent of the operator $T$.
The properties of the operator exponential and fractional powers of a $G$-sectorial operator are analysed alongside the question of the unique solubility of the Cauchy problem for the linear differential operator with $G$-sectorial operator-valued coefficient.
Bibliography: 8 titles.

DOI: https://doi.org/10.4213/sm1591

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English version:
Sbornik: Mathematics, 2006, 197:7, 977–995

Bibliographic databases:

UDC: 517.98
MSC: 47Bxx

Citation: M. F. Gorodnii, A. V. Chaikovskii, “A generalization of the concept of sectorial operator”, Mat. Sb., 197:7 (2006), 29–46; Sb. Math., 197:7 (2006), 977–995

Citation in format AMSBIB
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• https://doi.org/10.4213/sm1591
• http://mi.mathnet.ru/eng/msb/v197/i7/p29

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This publication is cited in the following articles:
1. Kasyanov P.O., Mel'nik V.S., Toscano S., “Solutions of Cauchy and periodic problems for evolution inclusions with multi-valued $w_{\lambda_0}$-pseudomonotone maps”, J. Differential Equations, 249:6 (2010), 1258–1287
2. Chaikovs'kyi A.V., “Cauchy problem for a nonlinear differential equation with $G$-sectorial operator coefficient”, Nonlinear Oscillations, 14:1 (2011), 114–125
3. Il'chenko Yu.V., Chaikovs'kyi A.V., “Cauchy Problem for a Differential Equation in the Banach Space with Generalized Strongly Positive Operator Coefficient”, Ukr. Math. J., 63:8 (2012), 1213–1233
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