Invariant diagonal operator ranges (Q1174974)

If \(T\) is a Hilbert-space operator, \({\mathcal A}\) the strongly closed operator algebra generated by \(T\) and for some vector \(x\), \({\mathcal A}x\) is the whole space then \(T\) is called strictly cyclic. If \(Q\) is a strictly cyclic weighted shift and \(D\) a diagonal operator the author characterizes in terms of the weights of \(Q\) and the diagonal entries of \(D\) the situation when the range of \(D\) belongs to some special lattice of operator ranges invarated by the commutant of \(Q\).