| A diagonal operator acting on the space H(B (0, R)) of functions analytic on the disk B(0, R) where 0 < R ≤ infinity is defined to be any continuous linear map on H( B(0, R)) having the monomials zn as eigenvectors. In this dissertation, examples of diagonal operators D acting on the spaces H( B(0, R)) where 0 < R < infinity, are constructed which fail to admit spectral synthesis; that is, which have invariant subspaces that are not spanned by collections of eigenvectors. Examples include diagonal operators whose eigenvalues are the points {n ae2piij/b : 0 ≤ j < b} lying on finitely many rays for suitably chosen a ∈ (0, 1) and b ∈ N , and more generally whose eigenvalues are the integer lattice points ZxiZ . Conditions for removing or perturbing countably many eigenvalues of a non-synthetic operator which yield another non-synthetic operator are also given. In addition, sufficient conditions are given for a diagonal operator on the space H(B(0, R)) of entire functions (for which R = infinity) to admit spectral synthesis. |
