Über das Reduktionsverfahren für diskrete Wiener-Hopf-Gleichungen mit unstetigem Symbol

Abstract

The results of the paper complete and improve some hitherto existing results about the reduction method for discrete Wiener-Hopf equations with discontinuous symbol and, on the other hand, illustrate some new aspects of this set of problems. Thus, it is proved that if the reduction method in $l^p(1\le p<\infty )$ is applicable to the Toeplitz operators $T(a_r),a_r\in L^{\infty }(r=1,\dots ,R)$ and if $a_1,\dots ,a_R$ have no common singularities (discontinuities), then it is applicable to $T(a_1\dots a_R)$. Furthermore, there is proved that for a large class of piecewise continuous symbols the reduction method in $L^p$ is applicable to $T(a)$ if and only if both $T(a)$ and $T(\bar{a})$ are invertible.