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

by Deters, Ian Nathaniel

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.