Criteria for normal solvability of systems of singular integral equations and Wiener--Hopf equations

J. Laiterer

Sbornik. Mathematics, Tome 12 (1970) no. 3, pp. 387-403

Résumé

Let $\Gamma$ be the unit circle and let $L^k$ ($k=1,2,\dots$) be the Hilbert space of vector functions $f(\zeta)=\{f_j(\zeta)\}_{j=1}^k$ with coordinates in $L_2(\Gamma)$. Theorem. {\it Let $a(\zeta),b(\zeta)$ $(\zeta\in\Gamma)$ be $m\times n$ matrices with elements continuous on $\Gamma$. In order for the singular integral operator $T,$ from $L^n$ to $L^m,$ $$ (Tf)(\zeta)=c(\zeta)f(\zeta)+\frac{d(\zeta)}{\pi i}\int_\Gamma\frac{f(z)}{z-\zeta}\,dz\qquad(f\in L^n) $$ to be normally solvable it is necessary and sufficient for the following two conditions to be satisfied}. a) The rank of each of the matrices $c(\zeta)+d(\zeta)$ and $c(\zeta)-d(\zeta)$ is independent of $\zeta$ on the unit circumference. b) {\it$\inf_{x\in(\operatorname{Ker}\,T)^\perp,\,\|x\|=1}\{\rho(Px,\operatorname{Ker}aI)+\rho(Qx,\operatorname{Ker}bI)\}>0.$} By $P$ we denote the orthogonal projector in $L^n$ defined by $(Pf)(\zeta)=\frac12f(\zeta)+\frac1{2\pi i}\int_\Gamma\frac{f(z)}{z-\zeta}\,dz$ ($f\in L^n$), $Q=I-P$. The conditions a) and b) are independent. The theorem is applicable to equations of Wiener–Hopf type. Bibliography: 11 titles.