Some estimates for the partial indices of measurable matrix-valued functions

I. M. Spitkovsky

Sbornik. Mathematics, Tome 39 (1981) no. 2, pp. 207-226

Résumé

Tests are given for nonnegativity, nonpositivity, and stability of partial indices of measurable bounded $n\times n$ matrix-valued functions defined on a contour $\Gamma$ along which the operator $S$ of singular integration is bounded in the spaces $L_p$, $1$. In particular, a sufficient condition is given for the coincidence of the partial indices of a matrix-valued function $G$ formulated in terms of the Hausdorff set of the matrices $G(t)$, $t\in \Gamma$. As auxiliary results, necessary and sufficient conditions are given for the operators of the form $T_G=\frac12(I-S)|\operatorname{Im}(I-S)$ to be Fredholm, or $n$- or $d$-normal in the case $G\in E^\pm_\infty+C$, and the behavior of the factorization is studied under the multiplication by such matrix-valued functions $G$ ($E^\pm_\infty$ are the Smirnov classes in the domains with boundary $\Gamma$ and $C$ is the class of functions continuous on $\Gamma$). In the case where $\Gamma$ is the unit circle, for the factorization in $L_2$ necessary and sufficient conditions are found for the nonnegativity (nonpositivity, and so on) of the partial indices. For a Lyapunov contour $\Gamma$ a sufficient condition (which is also necessary for $p=2$) is formulated for the vectorial boundary value problem of Riemann to be Fredholm in the spaces $L^n_p$ and $L^n_q$ ($q=p/(p-1)$). Bibliography: 38 titles.