Title

On The Cyclicity And Synthesis Of Diagonal Operators On The Space Of Functions Analytic On A Disk

Date of Award

2009

Document Type

Dissertation

Degree Name

Doctor of Philosophy (Ph.D.)

Department

Mathematics

First Advisor

Steven Seubert, Ph.D.

Second Advisor

Neal Carothers, Ph.D. (Committee Member)

Third Advisor

Sachi Sakthivel, Ph.D. (Committee Member)

Fourth Advisor

J. Gordan Wade, Ph.D. (Committee Member)

Abstract

A diagonal operator on the space of functions holomorphic on a disk of finite radius is a continuous linear operator having the monomials as eigenvectors. In this dissertation, necessary and sufficient conditions are given for a diagonal operator to be cyclic. Necessary and sufficient conditions are also given for a cyclic diagonal operator to admit spectral synthesis, that is, to have as closed invariant subspaces only the closed linear span of sets of eigenvectors. In particular, it is shown that a cyclic diagonal operator admits synthesis if and only if one vector, not depending on the operator, is cyclic. It is also shown that this is equivalent to existence of sequences of polynomials which seperate and have minimum growth on the eigenvalues of the operator.