Sectorial classes of inverse Stieltjes functions and L-systems

Abstract

We introduce sectorial classes of inverse Stieltjes functions actingon a finite-dimensional Hilbert space as well as scalar classes ofinverse Stieltjes functions based upon their limit behavior at minusinfinity and at zero. It is shown that a function from theseclasses can be realized as the impedance function of a singularL-system and the operator $\tilde A$ in a rigged Hilbert spaceassociated with the realizing system is sectorial. Moreover, it isestablished that the knowledge of the limit values of the scalarimpedance function allows to find an angle of sectoriality of theoperator $\tilde A$ as well as the exact angle of sectoriality of theaccretive main operator $T$ of such a system. The corresponding newformulas connecting the limit values of the impedance function andthe angle of sectoriality of $\tilde A$ are provided. Application ofthese formulas yields that the exact angle of sectoriality ofoperators $\tilde A$ and $T$ is the same if and only if the limitvalue at zero of the corresponding impedance function (along thenegative $x$-axis) is equal to zero. Examples of the realizingL-systems based upon the Schrodinger operator on half-line arepresented.