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# 数学代写|拓扑学代写TOPOLOGY代考|MATH067 TOPOLOGICAL DIVISORS OF ZERO

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## 数学代写|拓扑学代写TOPOLOGY代考|TOPOLOGICAL DIVISORS OF ZERO

An element $z$ in our Banach algebra $A$ is called a topological divisor of zero if there exists a sequence $\left{z_n\right}$ in $A$ such that $\left|z_n\right|=1$ and either $z z_n \rightarrow 0$ or $z_n z \rightarrow 0$. It is clear that every divisor of zero is also a topological divisor of zero. We denote the set of all topological divisors of zero by $Z$.
Theorem A. $Z$ is a subset of $S$.
Proof. Let $z$ be an element of $Z$ and $\left{z_n\right}$ a sequence such that $\left|z_n\right|=1$ and (say) $z z_n \rightarrow 0$. If $z$ were in $G$, then by the joint continuity of multiplication we would have $z^{-1}\left(z z_n\right)=z_n \rightarrow 0$, contrary to $\left|z_n\right|=1$.

Our next theorem relates to the manner in which $Z$ is distributed within $S$.
Theorem B. The boundary of $S$ is a subset of $Z$.
Proof. Since $S$ is closed, its boundary consists of all points in $S$ which are limits of convergent sequences in $G$. We show that if $z$ is such a point, that is, if $z$ is in $S$ and there exists a sequence $\left{r_n\right}$ in $G$ such that $r_n \rightarrow z$, then 2 is in $Z$. First, we see from $r_n{ }^{-1} z-1=r_n{ }^{-1}\left(z-r_n\right)$ that the sequence $\left{r_n^{-1}\right}$ is unbounded; for otherwise, we would have
$$\left|r_n^{-1} z-1\right|<1$$
for some $n$, so that $r_n^{-1} z$, and therefore $z=r_n\left(r_n^{-1} z\right)$, would be regular. Since $\left{r_n^{-1}\right}$ is unbounded, we may assume that $\left|r_n^{-1}\right| \rightarrow \infty$. If $z_n$ is now defined by $z_n=r_n^{-1} /\left|r_n^{-1}\right|$, then our conclusion follows from the observations that $\left|z_n\right|=1$ and
$$z z_n=\frac{z r_n^{-1}}{\left|r_n^{-1}\right|}=\frac{1+\left(z-r_n\right) r_n^{-1}}{\left|r_n^{-1}\right|}=\frac{1}{\left|r_n^{-1}\right|}+\left(z-r_n\right) z_n \rightarrow 0$$

## 数学代写|拓扑学代写TOPOLOGY代考|THE SPECTRUM

Let $T$ be an operator on a non-trivial Hilbert space. In the previous chapter, we defined the spectrum of $T$ to be the set
$$\sigma(T)={\lambda: T \text { – }-\lambda I \text { is singular }},$$
and we devoted a good deal of attention to the geometric ideas leading to this concept. We found-at least in the finite-dimensional case – that a number in $\sigma(T)$ is a value assumed by $T$, in the sense that $T$ acts on some non-zero vector as if it were scalar multiplication by that number. We shall see later that this formulation of the meaning of the spectrum has a much wider significance than we might at first suspect.

Let us now consider an element $x$ in our general Banach algebra $A$. By analogy with the above, we define the spectrum of $x$ to be the following subset of the complex plane:
$$\sigma(x)={\lambda: x-\lambda 1 \text { is singular }} .$$

## 数学代写|拓扑学代写TOPOLOGY代考|ORTHOGONAL COMPLEMENTS

$$x \perp y \Rightarrow|x+y|^2=|x-y|^2=|x|^2+|y|^2 .$$

$$0 \perp=H ; H \perp=0 ; S \cap H^{\perp} \subseteq 0 ; S_1 \subseteq S_2 \Rightarrow S_1 \perp \supseteq S_2{ }^{\perp}$$
$S^{\perp}$ 是一个封闭的线性子空间 $H$.

## 数学代写|拓扑学代写TOPOLOGY代考|ORTHONORMAL SETS

《left 缺少或无法识别的分隔符

(1) $i \neq j \Rightarrow e_i \perp e_j$
(2) $\left|e_i\right|=1$ 每一个 $i$.

## MATLAB代写

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