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Special factorization of a non-invertible integral Fredholm
operator of the second kind with
Hilbert–Schmidt kernel
G. A. Grigoryan Institute of Mathematics, National Academy of Sciences of Armenia
Abstract:
The problem of the special factorization of a non-invertible integral Fredholm
operator
$I-K$ of the second kind with Hilbert–Schmidt kernel is considered.
Here
$I$ is the identity operator and
$K$ is an integral operator:
$$
(Kf)(x)\equiv\int_0^1 \mathrm K(x,t)f(t)\,dt,
\qquad
f \in L_2[0,1].
$$
It is proved that
$\lambda=1$ is an eigenvalue of
$K$ of multiplicity
$n\geqslant1$ if and only if
$I-K=W_{+,1}\circ\dots\circ W_{+,n}\circ (I-K_n)\circ
W_{-,1}\circ\dots\circ W_{-,n}$, where the
$W_{+,j}$,
$W_{-,j}$,
$j=1,\dots,n$, are bounded operators in
$L_2[0,1]$ of a special structure
that are invertible from the left and the right, respectively.
Bibliography: 7 titles.
UDC:
517.968
MSC: 47G10,
47A68 Received: 04.07.2005 and 02.08.2006
DOI:
10.4213/sm1110
Fulltext:
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