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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Some operator-theoretic calculus for positive definite kernels


Author: Ameer Athavale
Journal: Proc. Amer. Math. Soc. 112 (1991), 701-708
MSC: Primary 47B38; Secondary 46E20, 47A57, 47B20, 47B37
MathSciNet review: 1068114
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Abstract: If $ \kappa $ is a positive definite kernel on the open unit disk $ D$ in the complex plane, then we associate with it a positive definite kernel $ \kappa '$ on $ D$ and correlate some operator theoretic properties of $ M\left( \kappa \right)$ and $ M\left( {\kappa '} \right)$, where $ M\left( \kappa \right)$ denotes the multiplication operator on the functional Hilbert space $ \mathcal{H}\left( \kappa \right)$ associated with $ \kappa $. The main emphasis of this paper is on the discussion of hyponormality and subnormality properties. We also construct a sequence of positive definite kernels $ {\kappa _{ - p}}\left( {p = 1,2, \ldots } \right)$ on $ D$ such that $ M\left( {{\kappa _{ - p}}} \right)$ is a $ \left( {p + 1} \right)$-isometry, but not a $ q$-isometry for any positive integer $ q$ less than or equal to $ p$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1991-1068114-8
PII: S 0002-9939(1991)1068114-8
Keywords: Positive definite kernel, hyponormal, subnormal, $ p$-isometry
Article copyright: © Copyright 1991 American Mathematical Society