SOME OPERATOR THEORETIC CALCULUS FOR POSITIVE DEFINITE KERNELS

If kappa is a positive definitie 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(kappa) and M(kappa'), where M(kappa) denotes the multiplication operator on the functional Hilbert space H (kappa) 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 (p = 1, 2, ...) on D such that M(kappa-p) is a (p + 1)-isometry, but not a q-isometry for any positive interger q less than or equal to p.