Examples of Diagonal Operators That Fail Spectral Synthesis on Spaces of Analytic Functions

The following dissertation is a discussion regarding a specific class of operators acting on spaces of functions analytic in a region. Both the space of functions analytic in the entire complex plane and the space of functions analytic in the unit disk are complete metrizable topological vector spaces. Then a diagonal operator D on either of these spaces is defined to be a continuous linear map, sending the space into itself, that has the monomials as its eigenvectors. If we let {λ_n} be the eigenvalues corresponding to these eigenvectors, and if we write an function f in terms of a power series f(z)=∑^∞_n=0a_nzⁿ, then we observe D(∑^∞_n=0a_nzⁿ)=∑^∞_n=0a_nλ_nzⁿ.

A closed subspace M is invariant for D if Df is in M for all f in M. The study of invariant subspaces is a popular topic in modern operator theory. We observe that the closed linear span of the orbit is the smallest closed invariant subspace for D containing f. If every invariant subspace for a diagonal operator D on a space of analytic functions can be expressed as a closed linear span of some subset of the eigenvectors of D, we say that D admits spectral synthesis. Until recently, it was not known whether or not every diagonal operator on the space of functions analytic on a disk admitted spectral synthesis. The dissertation of K. Overmoyer gave the first examples of nonsynthetic operators on this space. The present dissertation provides two new classes of operators which also fail synthesis on the space of functions analytic on the unit disk. It also provides the first known class of operators which fail spectral synthesis on the space of entire functions.