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Setting and solving several factorization problems for integral operators
N. B. Engibaryan Byurakan Astrophysical Observatory, National Academy of Sciences of Armenia
Abstract:
The problem of factorization
$$
I-K=(I-U_-)(I-U_+),
$$
is considered. Here
$I$ is the identity operator,
$K$ is a fixed integral operator of Fredholm type:
$$
(Kf)(x)=\int_a^bk(x,t)f(t)\,dt, \qquad -\infty\leqslant a<b\leqslant+\infty,
$$
$U_\pm$ are unknown upper and lower Volterra operators. Classes of generalized Volterra operators
$U_\pm$ are introduced such that
$I-U_\pm$ are not necessarily invertible operators in the spaces of functions on
$(a,b)$ under consideration. A combination of the method of non-linear factorization equations and a priori estimates brings forth new results on the existence and properties of the solution to this problem for
$k\geqslant 0$, both in the subcritical case
$\mu<1$ and in the critical case
$\mu=1$, where
$\mu=r(K)$ the spectral radius of the operator
$K$. In addition, the problem of non-Volterra factorization is posed and studied, when the kernels of
$U_+$ and
$U_-$ vanish on some parts
$S_-$ and
$S_+$ of the domain
$S=(a,b)^2$ such that
$S_+\cup S_-=S$.
UDC:
517.9
MSC: 45B05,
45D05,
47Gxx Received: 27.04.1999
DOI:
10.4213/sm529
Fulltext:
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