Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)
 Volume 16 (2021), 339 -- 359

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POSITIVE DEFINITENESS: FROM SCALAR TO OPERATOR-VALUED KERNELS

V. A. Menegatto

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.

2020 Mathematics Subject Classification: 43A35; 42A82.
Keywords: positive definiteness, strict positive definiteness, scalar kernels, matrix-valued kernels, operator-valued kernels.

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V. A. Menegatto
Departamento de Matemática -ICMC
Universidade de São Paulo
13560-970 São Carlos SP
Brazil

Current Affiliation:
Departamento de Matemática - ICE
Universidade Federal de Juiz de Fora
36036-330 Juiz de Fora MG
Brazil

e-mail: menegatt@gmail.com

http://www.utgjiu.ro/math/sma