Some operator-theoretic calculus for positive definite kernels

Abstract

If κ \kappa is a positive definite kernel on the open unit disk D D in the complex plane, then we associate with it a positive definite kernel κ ′ \kappa ’ on D D and correlate some operator theoretic properties of M ( κ ) M\left ( \kappa \right ) and M ( κ ′ ) M\left ( {\kappa ’} \right ) , where M ( κ ) M\left ( \kappa \right ) denotes the multiplication operator on the functional Hilbert space H ( κ ) \mathcal {H}\left ( \kappa \right ) associated with κ \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 κ − p ( p = 1 , 2 , … ) {\kappa _{ - p}}\left ( {p = 1,2, \ldots } \right ) on D D such that M ( κ − p ) M\left ( {{\kappa _{ - p}}} \right ) is a ( p + 1 ) \left ( {p + 1} \right ) -isometry, but not a q q -isometry for any positive integer q q less than or equal to p p .