Functions with Pick matrices having bounded number of negative eigenvalues

Abstract

A class is studied of complex valued functions defined on the unit disk (with a possible exception of a discrete set) with the property that all their Pick matrices have not more than a prescribed number of negative eigenvalues. Functions in this class, known to appear as pseudomultipliers of the Hardy space, are characterized in several other ways. It turns out that a typical function in the class is meromorphic with a possible modification at a finite number of points, and total number of poles and of points of modification does not exceed the prescribed number of negative eigenvalues. Bounds are given for the number of points that generate Pick matrices that are needed to achieve the requisite number of negative eigenvalues. A result analogous to Hindmarsh's theorem is proved for functions in the class.