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

J. Laiterer


Abstract: 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.

UDC: 517.948.3

MSC: 45E10, 45P05, 47G10, 47A56

Received: 30.01.1970

 English version: Mathematics of the USSR-Sbornik, 1970, 12:3, 387–403

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