Positive definiteness: from scalar to operator-valued kernels

Abstract

In this paper we present a short overview of results that provide relationships among scalar, matrix-valued and certain operator-valued positive definite kernels. We refine and extend some of them in order that they may be applied for strict positive definiteness as well. This is a topic not well explored in the literature but that has potential usefulness in the characterization of several classes of positive definite and strictly positive definite kernels. This is ratified in the paper with the inclusion of a number of applications and examples.

Keywords

• positive definiteness
• strict positive definiteness
• scalar kernels
• matrix-valued kernels
• operator-valued kernels