Let H be a separable Hilbert space and {e_n : n belong to Z} be an orthonormal basis in H. A bounded operator T is called the slant Toeplitz operator if <T ej , ei> =c2i−j , where c_n is the n-th Fourier series of a bounded Lebesgue measurable function on the unit circle T = {z belong to C : |z| = 1}. It has been shown [7] that T* is an isometry if and only if |fi(z)|^2 +|fi(−z)|^2 = 2 a.e. on T and if this is the case and fi belong to C(T), then either T is unitarily equivalent to a shift or to the direct sum of a shift and a rank one unitary, with infinite multiplicity (for the shift part, that is). Moreover, with some additional assumption on the smoothness and the zeros of fi, T* is similar to either the constant multiple of a shift or to the constant multiple of the direct sum of a shift and a rank one unitary, with infinite multiplicity. On the other hand, according to the terminologies in [10], an operator A that is constant with respect to a shift S if AS = SA and A S = SA . Therefore, in this article, we will study the operators that is constant with respect to T , i.e., bounded operator A satisfying AT = T A and A T = T A .

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 (b$_{i,j}$) be a bounded matrix on extit{ l}$^{2}$, $Bbb T={zinBbb C:|z|=1}$, and A be a bounded matrix on L$^{ 2}(mathbb{T)}$ satisfying the conditions 1.$langle Az^{2j},z^{2i} angle =sigma ^{-1}b_{ij}+|alpha |^{2}sigma ^{-1}langle Az^{j},z^{i} angle $; 2.$langle Az^{2j},z^{2i-1} angle =-alpha sigma ^{-1}b_{ij}+alpha sigma ^{-1}langle Az^{j},z^{i} angle $; 3.$langle Az^{2j-1},z^{2i} angle =-overline{alpha }sigma ^{-1}b_{ij}+overline{alpha }sigma ^{-1}langle Az^{j},z^{i} angle$; 4.$langle Az^{2j-1},z^{2i-1} angle =|alpha |^{2}sigma ^{-1}b_{ij}+sigma ^{-1}langle Az^{j},z^{i} angle $ hspace{-0.76cm} for all $i,jin mathbb{Z}$, where $sigma =1+|alpha |^{2},,alpha in mathbb{C},alpha eq0$. The above conditions evidently suggests that there is a "dyadic" relation in the entries of $A$. Here in the following picture illustrates how each $ij-$th entry of $A$ generates the 2 by 2 block in $A$ with entries ${a_{2i 2j}, a_{2i-1 2j}, a_{2i 2j-1}, a_{2i-1 2j-1}}.$ vspace{-0.3cm} egin{figure}[hp] egin{center} includegraphics[scale=0.42]{cubic.pdf} end{center} vspace{-0.8cm}caption{The dyadic recurrent form} end{figure} It has been shown [2] that $displaystyle A=sum_{n=0}^{infty }S^{n}BS^{ast n}$, where $Sz^i=sigma ^{-1/2}(overline{alpha }z^{2i}+z^{2i-1})$ and $$B=sumlimits_{i=-infty}^infty sumlimits_{j=-infty}^infty b_{ij}(u_{i}otimes u_{j}), u_{i}(z)=sigma ^{-1/2}z^{2i-1}(alpha -z).$$ In this paper, we shall use the above relations to compute $langle a_{i,j} angle $ explicitly. ewline Key words: shift operator, bounded matrix, dyadic recurrent formula, slant Toeplitz operator, separable Hilbert space 2.$langle Az^{2j},z^{2i-1} angle =-alpha sigma ^{-1}b_{ij}+alpha sigma ^{-1}langle Az^{j},z^{i} angle $ 3.$langle Az^{2j-1},z^{2i} angle =-overline{alpha }sigma ^{-1}b_{ij}+overline{alpha }sigma ^{-1}langle Az^{j},z^{i} angle $ 4.$langle Az^{2j-1},z^{2i-1} angle =|alpha |^{2}sigma ^{-1}b_{ij}+sigma ^{-1}langle Az^{j},z^{i} angle $ for all $i,jin mathbb{Z}$, where $sigma =1+|alpha |^{2},,alpha in mathbb{C},alpha eq0$ egin{figure}[hp] egin{center} includegraphics[scale=0.42]{cubic.pdf} end{center} caption{The dyadic recurrent form} end{figure} Since it has been shown [2] that $displaystyle A=sum_{n=0}^{infty }S^{n}BS^{ast n}$, where $ Sz^i=sigma ^{-1/2}(overline{alpha }z^{2i}+z^{2i-1})$ $ B=sum sum b_{ij}(u_{i}otimes u_{j})$ ;;; which $u_{i}(z)=sigma ^{-1/2}z^{2i-1}(alpha -z)$ Then we can use it to compute $langle Az^{j},z^{i} angle $ explicity if A satisfies the previous condition. ewline Key words: shift operator, bounded matrix, dyadic recurrent formula, slant Toeplitz operator, separable Hilbert space