Special factorization of a non-invertible integral Fredholm operator of the second kind with Hilbert–Schmidt kernel

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.
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