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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
$$
S_{a,\varphi}:=\{z\in\mathbb C\mid z\ne a,\ \lvert\arg(z-a)\rvert<\varphi\}
$$
and
$$
\text{there exists } M>0\quad \text{such that }
\|R_\lambda(T)\|\le MG(|\lambda-a|)\text{ for }\lambda\notin
S_{a,\varphi},
$$
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.
UDC:
517.98
MSC: 47Bxx Received: 23.11.2004 and 17.03.2006
DOI:
10.4213/sm1591
Fulltext:
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