Kernel dilation in reproducing kernel Hilbert space and its application to moment problems

Gary Doyle Faulkner and Ronald Wesley Shonkwiler

Vol. 77 (1978), No. 1, 103–115

DOI: 10.2140/pjm.1978.77.103

Abstract

We give a reformulation of Nagy’s Principal Theorem in terms of a dilation of a family of operators in reproducing kernel Hilbert space. In this setting we are able to generalize Nagy’s result to obtain dilations K of reproducing kernels K derived from certain families of operators. We define the concept of positive type for kernels K whose values are unbounded operators on a Hilbert space. The construction of K is such that it possesses a property, which we call splitting, not enjoyed by K. We show that the splitting property constitutes the utility of dilation theory and use it to solve moment problems.

Mathematical Subject Classification 2000

Primary: 46E20

Secondary: 44A60, 47A20, 47B50

Milestones

Received: 21 April 1977

Revised: 18 December 1977

Published: 1 July 1978