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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
DOI: https://doi.org/10.1090/S0002-9939-1991-1068114-8
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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Keywords: Positive definite kernel, hyponormal, subnormal, <IMG WIDTH="16" HEIGHT="37" ALIGN="MIDDLE" BORDER="0" SRC="images/img17.gif" ALT="$p$">-isometry
Article copyright: © Copyright 1991 American Mathematical Society