Rotationally invariant kernel systems of analytic functions on the unit disk

dc.creatorDennis, Thomas Burrows
dc.date.accessioned2022-06-22T18:28:06Z
dc.date.available2022-06-22T18:28:06Z
dc.date.copyright1977
dc.date.issued1977
dc.description.abstractThis 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 K[lowered proptional to](s,t) is [proptional to](st*) 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.
dc.description.departmentMathematics, Department of
dc.format.digitalOriginreformatted digital
dc.format.mimetypeapplication/pdf
dc.identifier.other3340841
dc.identifier.urihttps://hdl.handle.net/10657/9823
dc.language.isoen
dc.rightsThis item is protected by copyright but is made available here under a claim of fair use (17 U.S.C. Section 107) for non-profit research and educational purposes. Users of this work assume the responsibility for determining copyright status prior to reusing, publishing, or reproducing this item for purposes other than what is allowed by fair use or other copyright exemptions. Any reuse of this item in excess of fair use or other copyright exemptions requires express permission of the copyright holder.
dc.titleRotationally invariant kernel systems of analytic functions on the unit disk
dc.type.dcmiText
dc.type.genreThesis
dcterms.accessRightsThe full text of this item is not available at this time because it contains documents that are presumed to be under copyright and are accessible only to users who have an active CougarNet ID. This item will continue to be made available through interlibrary loan.
thesis.degree.collegeCollege of Natural Sciences and Mathematics
thesis.degree.departmentMathematics, Department of
thesis.degree.disciplineMathematics
thesis.degree.grantorUniversity of Houston
thesis.degree.levelDoctoral
thesis.degree.nameDoctor of Philosophy

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