Rotationally invariant kernel systems of analytic functions on the unit disk



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This paper investigates the kernel systems, {K[lowered proportional to],U ,H [lowered proportional to], Q[lowered proportional to]} relatively to a complete inner product space {Y,<•,•>}, which have the property that Klowered proptional to is proptional to for some operator valued analytic function [proportional to] on the unit disk U. The connecting link between the set U, the Hilbert space {H[lowered proportional to], Q[lowered proportional to]} of functions from U to Y, and the kernel in such a system is the recipe <f(t),[Eta]> = Q[lowere proportional to](f, K[lowered proportional to](•, t)[Eta]), for f in H[lowered proportional to], t in U, and [Eta] in Y. Characterizations are given of the analytic functions [proportional to] which determine the kernels, the spaces H[lowered proportional to], and the inner products Q[lowered proportional to]. Next, there is a characterization of the systems which possess this type of kernel. The latter characterization shows that these systems are the ones in which the associated Hilbert spaces are rotationally Invariant. This concept of viewing the rotations of the disk as linear operators on such Hilbert spaces leads to an investigation of the operators potentially definable by composition with other reversible analytic functions from the disk onto itself. A characterization is given of the kernel systems which admit one of these operators, and when this characterization is applied to a rotationally invariant system, it shows that either all such operators are admissible or else none of these operators except for those defined by rotations are continuous. Finally, the rotationally invariant systems which admit this entire group of operators are characterized.