Setting and solving several factorization problems for integral operators

N. B. Engibaryan

Sbornik. Mathematics, Tome 191 (2000) no. 12, pp. 1809-1825

Résumé

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: $$ (Kf)(x)=\int_a^bk(x,t)f(t)\,dt, \qquad -\infty\leqslant a\leqslant+\infty, $$ $U_\pm$ 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 $\mu1$ 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 $S_+\cup S_-=S$.