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Kolmogorov decomposition of conditionally completely positive definite kernels

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In this paper, we investigate conditionally completely positive definite kernels in the setting of Hilbert \(C^*\)-modules. At first, we present a correspondence between conditionally completely positive definite kernels and completely positive definite kernels. Utilizing this relation, we give the Kolmogorov decomposition for a certain conditionally completely positive definite kernel. We present a characterization of conditionally completely positive definite kernels majorized by a kernel under some mild conditions. Finally, as an application of this decomposition, we find a sufficient condition for the existence of an extreme point of a convex set consisting of some special kernels.

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Authors and Affiliations

  1. Department of Pure Mathematics, Ferdowsi University of Mashhad, P. O. Box 1159, Mashhad, 91775, Iran

    Mostafa Ghaemi

  2. Department of Pure Mathematics, Center of Excellence in Analysis on Algebraic Structures (CEAAS), Ferdowsi University of Mashhad, P. O. Box 1159, Mashhad, 91775, Iran

    M. S. Moslehian

  3. Department of Mathematics, Shanghai Normal University, Shanghai, 200234, People’s Republic of China

    Qingxiang Xu

Authors
  1. Mostafa Ghaemi
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  2. M. S. Moslehian
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  3. Qingxiang Xu
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Correspondence to M. S. Moslehian.

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Ghaemi, M., Moslehian, M.S. & Xu, Q. Kolmogorov decomposition of conditionally completely positive definite kernels. Positivity 25, 515–530 (2021). https://doi.org/10.1007/s11117-020-00777-3

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