Geodesic
Special factorization of a non-invertible integral Fredholm
Sbornik. Mathematics, Tome 198 (2007) no. 5, pp. 627-637 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
@article{SM_2007_198_5_a1,
     author = {G. A. Grigoryan},
     title = {Special factorization of a~non-invertible integral {Fredholm}},
     journal = {Sbornik. Mathematics},
     pages = {627--637},
     year = {2007},
     volume = {198},
     number = {5},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2007_198_5_a1/}
}
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G. A. Grigoryan. Special factorization of a non-invertible integral Fredholm. Sbornik. Mathematics, Tome 198 (2007) no. 5, pp. 627-637. http://geodesic.mathdoc.fr/item/SM_2007_198_5_a1/

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