在探討與斜托普利茲算子可交換的有界算子中，發展出一種二進位遞迴方程組，此方程組的解和一個算子方程式的有界解A 是有密切相關的，此算子方程式為
ϕ(A) = S∗AS = λA + B
，其中B 是一個已知的算子，λ是一個複數，S 是由Tζ+ξz所給定的一個移算子。在這篇碩士論文主要探討的是如何把此算子方程式對應小於或等於1 的特徵值的特徵向量用二進位遞迴方程組表示出來。

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.

1 Introduction 1
2 Some preliminary facts 4
3 Eigenvectors for the action 6
References 13

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