Non-Synthetic diagonal operators on the space of functions analytic on the unit disk

Abstract

Examples are given of continuous operators of functions analytic on the unit disk having the monomials as eigenvectors which fail spectral synthesis (that is, which have closed invariant subspaces which are not the closed linear span of collections of eigenvectors). Examples include the diagonal operator having as eigenvalues an enumeration of $\IL\equiv \{m+in: m,n\in\mathbb{Z}\}$ and diagonal operators having as eigenvalues enumerations of $\{n^{1/p}e^{2\pi ij/3p}: 0\leq j \lt p\}$ where $p$ is an integer at least~2.