Möbius functions of the shift and extremal operators (Q1911418)

As a continuation of the author's previous work, this paper deals with matrix representations of the extremal operator \(S^+|\text{Ker } P(S^+)\). The motivation is in the convergence theory of the iteration process \(y_{s + 1} = Ay_s + b\). Based on some natural identities involving Möbius functions of the shift operator on the Hardy space \(H^2\), an approach is proposed. The advantage of this new approach lies in the fact that it is simple and reduces the amount of computations. In fact, the operator identities are not without interest in their own right: they are valid for an isometry in an arbitrary Hilbert space.