2020/08/21 by Zenon Jan Jabłoński, Il Bong Jung, Jabłoński, Zenon Jan +3
Mathematics · #44A60 #47A20 #47A60 #47B20 #Algebraic and Geometric Analysis #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Spectral Theory in Mathematical Physics #math.FA #msc:44A60 #msc:47A20 #msc:47A60 #msc:47B20
paper · pdf · doi:10.48550/arxiv.2008.09494
openalex publication_date 2020/08/21 · arxiv created 2022/01/25 · arxiv updated 2022/01/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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∞(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).