Conditionally positive definiteness in operator theory

TL;DR

This paper thoroughly investigates conditionally positive definite operators, establishing criteria, representations, and functional calculus, and explores their properties, including closure under powers and applications to inequalities and subnormal contractions.

Contribution

It introduces new criteria and representations for conditionally positive definite operators, including a functional calculus, and explores their structural properties and applications.

Findings

Operators generating conditionally positive definite sequences include subnormal operators and isometries.

Criteria for positive definiteness of exponential growth sequences are established.

A new functional calculus for these operators is developed, with applications to inequalities.

Abstract

In this paper we extensively investigate the class of conditionally positive definite operators, namely operators generating conditionally positive definite sequences. This class itself contains subnormal operators, $2$- and $3$-isometries and much more beyond them. Quite a large part of the paper is devoted to the study of conditionally positive definite sequences of exponential growth with emphasis put on finding criteria for their positive definiteness, where both notions are understood in the semigroup sense. As a consequence, we obtain semispectral and dilation type representations for conditionally positive definite operators. We also show that the class of conditionally positive definite operators is closed under the operation of taking powers. On the basis of Agler's hereditary functional calculus, we build an $L^{\infty}(M)$-functional calculus for operators of this class, where $M$ is an associated semispectral measure. We provide a variety of applications of this calculus to inequalities involving polynomials and analytic functions. In addition, we derive new necessary and sufficient conditions for a conditionally positive definite operator to be a subnormal contraction (including a telescopic one).