令 $l^2(Bbb Z)$ 是建立在標準複數基底 ${e_n:ninBbb
Z}$, 中一希爾伯特空間的平方累加重序列，而我們考慮一個有界矩陣A在 $l^2(Bbb Z)$ 滿足下列方程組：
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}
對所有的 i, j，其中P = ( pij )，Q = ( qij )，V = ( vij )，W = ( wij ) 都是在 $l^2(Bbb Z)$ 的有界矩陣且a, b, c, d 。
很明顯地，我們將上述方程組的解分為幾組無限維矩陣，且每一組中的元素皆與“二進位”有關。而[4]這篇文章，說明了在此系統方程組中，建立可交換矩陣的過程似乎是很艱難的，而藉由移算子，我們化簡在 B(H) 上的算子方程式，透過Hardy算子分類理論，可以變得比較容易。這篇文章主要的目的是將這一系列的二進位遞迴方程組找到確切的齊次詳解。

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.

Introduction 1
The explicit solutions for phi(A)=la A 6
Bibliography 14

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14
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