Special factorization of a~non-invertible integral Fredholm

G. A. Grigoryan

Sbornik. Mathematics, Tome 198 (2007) no. 5, pp. 627-637

Résumé

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.