讓 T 是單位圓, (mu)是一個在T上的 Borel 機率測度,(phi)是一個在T上的 Lebesgue 可測函數. 在這篇論文中,我們考慮在 L^2(T,(mu))上的加權合成算子 W(phi)定義成W(phi)f:=(phi)*(f(circle)(tau)),f 屬於 L^2(T),
(tau)是這個映射:(tau)(z)=z^2, z 屬於 T. 當 W(phi)是保距並且 (mu)<< m 時, 我們將研讀 W(phi) 的von Neumann-Wold 分解, 無論 m 是正規化的 Lebesgue 測度.

Let T be the unit circle,(mu) be a Borel probability measure on T and (phi) be a bounded Lebesgue measurable function on T. in this paper we consider the weighted composition operator W(phi) on L^2(T,mu) defined by
W(phi)f:=(phi)*(f(circle)(tau)), f in L^2(T),
where (tau) is the map (tau)(z)=z^2, z in T.
We will study the von Neumann-Wold decomposition of W(phi) when W(phi) is an isometry and (mu)<< m,where m is the normalized Lebesgue measure on T.

1. INTRODUCTION..........................................................3
1.1. Absolutely Continuous Positive Measure........ .....................4
2. BOUNDEDNESS OF W(phi).................................................4
2.1. General Conditions for The Boundedness of W(phi)....................4
2.2. Boundedness of W(phi) and Eigenfunctions of A?......................5
2.3. Remark: On The Existence of g.......................................11
3. STRUCTURE OF W(phi)...................................................12
3.1. Von Neumann-Wold Decomposition of Isometry..........................12
3.2. Von Neumann-Wold Decomposition of W? on L^2(T)......................13
4. FINAL THOUGHTS........................................................16
References...............................................................17

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