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Conditional positive definiteness in operator theory

Tom 578 / 2022

Zenon Jan Jabłoński, Il Bong Jung, Jan Stochel Dissertationes Mathematicae 578 (2022), 1-64 MSC: Primary 47B20, 44A60; Secondary 47A20, 47A60. DOI: 10.4064/dm846-1-2022 Opublikowany online: 11 April 2022

Streszczenie

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, complete hypercontractions of order $2$ 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 condition).

Autorzy

  • Zenon Jan JabłońskiInstytut Matematyki
    Uniwersytet Jagielloński
    Łojasiewicza 6
    30-348 Kraków, Poland
    e-mail
  • Il Bong JungDepartment of Mathematics
    Kyungpook National University
    Daegu 41566, Korea
    e-mail
  • Jan StochelInstytut Matematyki
    Uniwersytet Jagielloński
    Łojasiewicza 6
    30-348 Kraków, Poland
    e-mail

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