Let $l^2(Bbb Z)$ be the Hilbert space of square summable double sequences of complex numbers with standard basis ${e_n:ninBbb Z}$, and let us consider a bounded matrix $A$ on $l^2(Bbb Z)$ satisfying the following system of equations egin{itemize} item[1.] $lan Ae_{2j},e_{2i} an=p_{ij}+alan Ae_{j},e_i an$; item[2.] $lan Ae_{2j},e_{2i-1} an=q_{ij}+blan Ae_{j},e_{i} an$; item[3.] $lan Ae_{2j-1},e_{2i} an=v_{ij}+clan Ae_{j},e_{i} an$; item[4.] $lan Ae_{2j-1},e_{2i-1} an=w_{ij}+dlan Ae_{j},e_{i} an$ end{itemize} for all $i,j$, where $P=(p_{ij})$, $Q=(q_{ij})$, $V=(v_{ij})$, $W=(w_{ij})$ are bounded matrices on $l^2(Bbb Z)$ and $a,b,c,dinBbb C$. This type dyadic recurrent system arises in the study of bounded operators commuting with the slant Toeplitz operators, i.e., the class of operators ${{cal T}_vp:vpin L^infty(Bbb T)}$ satisfying $lan {cal T}_vp e_j,e_i an=c_{2i-j}$, where $c_n$ is the $n$-th Fourier coefficient of $vp$. It is shown in [10] that the solutions of the above system are closely related to the bounded solution $A$ for the operator equation [ phi(A)=S^*AS=lambda A+B, ] where $B$ is fixed, $lambdainBbb C$ and $S$ the shift given by ${cal T}_{arzeta+arxi z}^*$ (with $zetaxi ot=0$ and $|zeta|^2+|xi|^2=1$). In this paper, we shall characterize the ``eigenvectors" for $phi$ for the eigenvalue $lambda$ with $|lambda|leq1$, in terms of dyadic recurrent systems similar to the one above.

Let $l^2(Bbb Z)$ be the Hilbert space of square summable double sequences of complex numbers with standard basis ${e_n:ninBbb Z}$, and let us consider a bounded matrix $A$ on $l^2(Bbb Z)$ satisfying the following system of equations egin{itemize} item[1.] $lan Ae_{2j},e_{2i} an=p_{ij}+alan Ae_{j},e_i an$; item[2.] $lan Ae_{2j},e_{2i-1} an=q_{ij}+blan Ae_{j},e_{i} an$; item[3.] $lan Ae_{2j-1},e_{2i} an=v_{ij}+clan Ae_{j},e_{i} an$; item[4.] $lan Ae_{2j-1},e_{2i-1} an=w_{ij}+dlan Ae_{j},e_{i} an$ end{itemize} for all $i,j$, where $P=(p_{ij})$, $Q=(q_{ij})$, $V=(v_{ij})$, $W=(w_{ij})$ are bounded matrices on $l^2(Bbb Z)$ and $a,b,c,dinBbb C$. par It is clear that the solutions of the above system of equations introduces a class of infinite matrices whose entries are related ``dyadically". In cite{Ho:g}, it is shown that the seemingly complicated task of constructing these matrices can be carried out alternatively in a systematical and relatively simple way by applying the theory of Hardy classes of operators through certain operator equation on ${cal B}({cal H})$ (space of bounded operators on $cal H$) induced by a shift. Our purpose here is to present explicit formula for the homogeneous solutions this equation.