Solutions to a dyadic recurrent system and a certain action on B(H) induced by shifts

It is clear that the solutions of the above system of equations introduces a new class of infinite matrices whose entries are related “dyadically”. In this paper, we will show that while the task of constructing these matrices explicitly using purely algebraic methods may appear to be very complicated and tedious, if not impossible, it can be carried out alternatively in a systematical and relatively simple way by applying the theory of Hardy classes of operators through a certain action on B(H) (space of bounded operators on H) induced by a shift.