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

Let l2(Z)l2(Z) be the Hilbert space of square summable double sequences of complex numbers with standard basis {en:n∈Z}{en:n∈Z}, and let us consider a bounded matrix AA on l2(Z)l2(Z) satisfying the following system of equations 1. 〈Ae2j,e2i〉=pij+a〈Aej,ei〉〈Ae2j,e2i〉=pij+a〈Aej,ei〉; 2. 〈Ae2j,e2i−1〉=qij+b〈Aej,ei〉〈Ae2j,e2i−1〉=qij+b〈Aej,ei〉; 3. 〈Ae2j−1,e2i〉=vij+c〈Aej,ei〉〈Ae2j−1,e2i〉=vij+c〈Aej,ei〉; 4. 〈Ae2j−1,e2i−1〉=wij+d〈Aej,ei〉〈Ae2j−1,e2i−1〉=wij+d〈Aej,ei〉 for all i,ji,j, where P=(pij)P=(pij), Q=(qij)Q=(qij), V=(vij)V=(vij), W=(wij)W=(wij) are bounded matrices on l2(Z)l2(Z) and a,b,c,d∈Ca,b,c,d∈C. 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)B(H) (space of bounded operators on HH) induced by a shift