Cyclicity of Adjoints
T = UU* is strongly *-cyclic, but neither T nor T* is cyclic. Are there common cyclic vectors for pure operators in P(S*)? Which operators T have the property that for every polynomial p, p(T) is cyclic? Which ones have the property that every pure operator in W(T) is cyclic? Does the first property imply the second? If A is a matrix, and p(A) is cyclic or p(A) = 0 for every polynomial p, then A is a 2 2 matrix with only one eigenvector!