令X 為一個緊緻Hausdorff 拓撲空間。C(X)為一個收集所有定義在X 上的復
數值連續函數的Banach 空間。如果在C(X)上的算子P 具有以下的性質，P 平方
等於P 而且對所有範數為1 的復數λ，P+λ(I-P)是一個保距映射，則稱P 為一
個一般化的bicircular 投影。
在此篇論文中，我們主要研究了兩個複合算子或兩個加權複合算子的平均為
C(X)上的投影。如果一個單位算子和一個複合算子的平均為C(X)上的投影，則
此投影自動變成一個一般化的bicircular 投影。並給予一個例子說明一個單位算
子和一個加權複合算子的平均為C(X)上的投影，但不是一個一般化的bicircular
投影。
我們也討論了兩個有界線性算子的平均是一個Banach 空間上的投影。而主要
的結果如下，令T1 和T2 是 Banach 空間上的兩個有界線性算子，而且Q 等於
T1 和T2 的平均，如果T1。T2 = T2。T1 和T1 = T2 = Id 那麼Q 就會是一個tripotent，
也就是說Q 的三次方會等於Q。

Let X be a compact Hausdorff topological space. The Banach space C(X) consists of all
continuous complex value functions with the supnorm. An operator P on C(X) is called a
generalized bicircular projection if P + λ(I − P) is an isometry for all |λ| = 1, λ in C and
P2 = P.
In this thesis, we study some projections which are the averages of two composition
operators or two weighted composition operators on C(X). If a projection is the average of
the identity and a composition operator, it is a generalized bicircular projection. And give
an example of a projection which is the average of the identity and a weighted composition
operator, but not a generalized bicircular projection.
We also discuss some projections which are the average of two bounded linear operators
on a Banach space. And the main result is that, let T1 and T2 are two bounded linear
operators on a Banach space, and Q = T1+T2
2 . If T1 。T2 = T2 。T1 and T2
1 = T2
2 = Id then Q
is a tripotent, i.e. Q3 = Q.

Contents
1 Introduction 4
2 When is the average of two composition operators on C(X) a projection ? 6
3 When is the average of two bounded linear operators a tripotent ? 15
References 20

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