Multiplication operators with deficiency indices (p,p) and sampling formulas in reproducing kernel Hilbert spaces of entire vector valued functions

Abstract

A number of recent papers have established connections between reproducing kernel Hilbert spaces H of entire functions, de Branges spaces, sampling formulas and a class of symmetric operators with deficiency indices (1,1). In this paper analogous connections between reproducing kernel Hilbert spaces of entire vector valued functions, de Branges spaces of entire vector valued functions, sampling formulas and symmetric operators with deficiency indices (p,p) are obtained. Enroute, an analog of L. de Branges' characterization of the reproducing kernel Hilbert spaces of entire functions that are now called de Branges spaces is obtained for the p×1 vector valued case. A special class of these de Branges spaces of p×1 vector valued entire functions is identified as a functional model for M. G. Krein's class of entire operators with deficiency indices (p,p).