# 数学代写|泛函分析作业代写Functional Analysis代考|Algebras of Normal Operators

## 数学代写|泛函分析作业代写Functional Analysis代考|Algebras of Normal Operators

The construction of the continuous functional calculus for normal operators is based on several lemmas. Assume throughout that $H$ is a nonzero complex Hilbert space and that $A_0 \in \mathcal{L}^c(H)$ is a normal operator. Let
$$\mathcal{A}_0 \subset \mathcal{L}^c(H)$$
be the smallest (unital) $\mathrm{C}^$ subalgebra that contains $A_0$. Lemma 5.66. $\mathcal{A}_0$ is commutative and every operator $A \in \mathcal{A}_0$ is normal. Moreover, if $B \in \mathcal{L}^c(H)$ satisfies $B A_0=A_0 B$ and $B A_0^=A_0^* B$, then $B$ commutes with every element of $\mathcal{A}_0$.
Proof. Define
$$\mathcal{B}:=\left{B \in \mathcal{L}^c(H) \mid A_0 B=B A_0 \text { and } B A_0^=A_0^ B\right} .$$
Then $\mathcal{B}$ is a closed subspace of $\mathcal{L}^c(H)$ that contains the identity and is invariant under composition. Moreover, $A_0 \in \mathcal{B}$ because $A_0$ and $A_0^$ commute, and $B \in \mathcal{B}$ implies $B^ \in \mathcal{B}$. Hence $\mathcal{B}$ is a $\mathrm{C}^$ subalgebra of $\mathcal{L}^c(H)$ that contains $A_0$. Hence the set $$\mathcal{C}:=\left{C \in \mathcal{L}^c(H) \mid B C=C B \text { for all } B \in \mathcal{B}\right}$$ is also a $\mathrm{C}^$ subalgebra of $\mathcal{L}^c(H)$ that contains $A_0$. Moreover, since $A_0, A_0^* \in \mathcal{B}$ we have $\mathcal{C} \subset \mathcal{B}$. Hence $\mathcal{C}$ is commutative, so every $C \in \mathcal{C}$ is normal. Since $\mathcal{C}$ is a $\mathrm{C}^*$ subalgebra of $\mathcal{L}^c(H)$ and $A_0 \in \mathcal{C}$, we have $\mathcal{A}_0 \subset \mathcal{C}$ and this proves Lemma 5.66.

Lemma 5.67. Let $\operatorname{Spec}\left(\mathcal{A}_0\right)$ be the set of maximal ideals in $\mathcal{A}_0$. Then, for every $A \in \mathcal{A}_0$, there exists a unique function $f_A: \operatorname{Spec}\left(\mathcal{A}_0\right) \rightarrow \mathbb{C}$ such that
$$f_A(\mathcal{J}) \mathbb{1}-A \in \mathcal{J}$$
for all $\mathcal{J} \in \operatorname{Spec}\left(\mathcal{A}_0\right)$. Equip $\operatorname{Spec}\left(\mathcal{A}_0\right)$ with the weakest topology such that $f_A$ is continuous for every $A \in \mathcal{A}_0$. Then $\operatorname{Spec}\left(\mathcal{A}_0\right)$ is a compact Hausdorff space, the Gelfand representation
$$\mathcal{A}_0 \rightarrow C\left(\operatorname{Spec}\left(\mathcal{A}_0\right)\right): A \mapsto f_A$$
is an isometric $C^*$ algebra isomorphism and
$$f_A\left(\operatorname{Spec}\left(\mathcal{A}_0\right)\right)=\sigma(A) \quad \text { for all } A \in \mathcal{A}_0$$

## 数学代写|泛函分析作业代写Functional Analysis代考|Spectral Measures

Assume that $H$ is a nonzero complex Hilbert space and $A \in \mathcal{L}^c(H)$ is a normal operator. Then the spectrum $\Sigma:=\sigma(A) \subset \mathbb{C}$ is a nonempty compact set of complex numbers by Theorem 5.44. Let
$$C(\Sigma) \rightarrow \mathcal{L}^c(H): f \mapsto f(A)$$
be the $\mathrm{C}^$ algebra homomorphism introduced in Theorem 5.70. The purpose of the present section is to assign to $A$ a Borel measure on $\Sigma$ with values in the space of orthogonal projections on $H$, called the spectral measure of $A$. When $A$ is a compact operator this measure assigns to each Borel set $\Omega \subset \Sigma$ the spectral projection $$P_{\Omega}:=\sum_{\lambda \in \sigma(A) \cap \Omega} P_\lambda$$ associated to all the eigenvalues of $A$ in $\Omega$ (see Remark 5.71). The general construction of the spectral measure is considerably more subtle and is closely related to an extension of the homomorphism in Theorem 5.70 to the $\mathrm{C}^$ algebra $B(\Sigma)$ of all bounded Borel measurable functions on $\Sigma$. The starting point for the construction of this extension and the spectral measure is the observation that every element $x \in H$ determines a conjugation equivariant bounded linear functional $\Lambda_x: C(\Sigma) \rightarrow \mathbb{C}$ via the formula
$$\Lambda_x(f):=\langle x, f(A) x\rangle \quad \text { for } f \in C(\Sigma) .$$
Since $\Lambda_x(\bar{f})=\overline{\Lambda_x(f)}$ for all $f \in C(\Sigma)$, the functional $\Lambda_x$ is uniquely determined by its restriction to the subspace $C(\Sigma, \mathbb{R})$ of real valued continuous functions. This restriction takes values in $\mathbb{R}$ and the restricted functional $\Lambda_x: C(\Sigma, \mathbb{R}) \rightarrow \mathbb{R}$ is positive by Theorem 5.70 , i.e. for all $f \in C(\Sigma, \mathbb{R})$,
$$f \geq 0 \quad \Longrightarrow \quad \Lambda_x(f) \geq 0 \text {. }$$
Hence the Riesz Representation Theorem asserts that $\Lambda_x$ can be represented by a Borel measure. Namely, let $\mathcal{B} \subset 2^{\Sigma}$ be the Borel $\sigma$-algebra. Then, for every $x \in \Sigma$, there exists a unique Borel measure $\mu_x: \mathcal{B} \rightarrow[0, \infty)$ such that
$$\int_{\Sigma} f d \mu_x=\langle x, f(A) x\rangle \quad \text { for all } f \in C(\Sigma, \mathbb{R}) \text {. }$$
(See [50, Cor 3.19].) These Borel measures can be used to define the desired extension of the $\mathrm{C}^*$ algebra homomorphism $C(\Sigma) \rightarrow \mathcal{L}^c(H)$ to $B(\Sigma)$ as well as the spectral measure of $A$.

# 泛函分析代考

## 数学代写|泛函分析作业代写Functional Analysis代考|Algebras of Normal Operators

$$\mathcal{A}0 \subset \mathcal{L}^c(H)$$ 成为最小的 (统一的) \mathrm{C}个 包含的子代数 $A_0$. 引理 5.66。 $\mathcal{A}_0$ 是可交换的，每 个运算符 $A \in \mathcal{A}_0$ 是正常的。此外，如果 $B \in \mathcal{L}^c(H)$ 满足 $B A_0=A_0 B$ 和 $B A_0^{=} A_0^* B$ ，然后 $B$ 通勤的每一个元素 $\mathcal{A}_0$. 证明。定义 $\backslash$ mathcal{B $}:=\backslash$ eft $\left{B \backslash\right.$ in $\backslash$ mathcal $\left{\langle}^{\wedge} c(H) \backslash\right.$ mid A $0 B=B A_{-} 0 \backslash$ text ${$ and $}$ B A_ $\left.0^{\wedge}=A_{-} 0^{\wedge} B \backslash r i g h t\right}$.

$\backslash$ mathcal ${C}:=\backslash$ left $\left{C \backslash\right.$ in $\backslash$ mathcal $\left{\langle}^{\wedge} C(H) \backslash m i d ~ B C=C B \backslash\right.$ text ${$ for all $}$ B $\backslash$ in $\backslash$ mathcal ${B} \backslash$ right $}$ 因此 $\mathcal{C}$ 是可交换的，所以每个 $C \in \mathcal{C}$ 是正常的。自从 $\mathcal{C}$ 是一个 $\mathrm{C}^$ 的子代数 $\mathcal{L}^c(H)$ 和 $A_0 \in \mathcal{C}$ ，我们有 $\mathcal{A}_0 \subset \mathcal{C}$ 这证明了引理 5.66。 引理 5.67。让 $\operatorname{Spec}\left(\mathcal{A}_0\right)$ 是最大理想的集合 $\mathcal{A}_0$. 然后，对于每一个 $A \in \mathcal{A}_0$ ，存在唯一函 数 $f_A: \operatorname{Spec}\left(\mathcal{A}_0\right) \rightarrow \mathbb{C}$ 这样 $$f_A(\mathcal{J}) 1-A \in \mathcal{J}$$ 对全部 $\mathcal{J} \in \operatorname{Spec}\left(\mathcal{A}_0\right)$. 装备 $\operatorname{Spec}\left(\mathcal{A}_0\right)$ 最弱的拓扑使得 $f_A$ 对每个都是连续的 $A \in \mathcal{A}_0$. 然后 $\operatorname{Spec}\left(\mathcal{A}_0\right)$ 是紧致的 Hausdorff空间，Gelfand 表示 $$\mathcal{A}_0 \rightarrow C\left(\operatorname{Spec}\left(\mathcal{A}_0\right)\right): A \mapsto f_A$$ 是等距的 $C^$ 代数同构和
$$f_A\left(\operatorname{Spec}\left(\mathcal{A}_0\right)\right)=\sigma(A) \quad \text { for all } A \in \mathcal{A}_0$$

## 数学代写|泛函分析作业代写Functional Analysis代考|Spectral Measures

$$C(\Sigma) \rightarrow \mathcal{L}^c(H): f \mapsto f(A)$$

$$P_{\Omega}:=\sum_{\lambda \in \sigma(A) \cap \Omega} P_\lambda$$

$$\Lambda_x(f):=\langle x, f(A) x\rangle \quad \text { for } f \in C(\Sigma) .$$

$$f \geq 0 \quad \Longrightarrow \quad \Lambda_x(f) \geq 0$$

$$\int_{\Sigma} f d \mu_x=\langle x, f(A) x\rangle \quad \text { for all } f \in C(\Sigma, \mathbb{R})$$
(参见 [50，Cor 3.19]。) 这些 Borel 测量可用于定义所需的扩展 $\mathrm{C}^*$ 代数同态 $C(\Sigma) \rightarrow \mathcal{L}^c(H)$ 到 $B(\Sigma)$ 以及光谱测量 $A$.

myassignments-help数学代考价格说明

1、客户需提供物理代考的网址，相关账户，以及课程名称，Textbook等相关资料~客服会根据作业数量和持续时间给您定价~使收费透明，让您清楚的知道您的钱花在什么地方。

2、数学代写一般每篇报价约为600—1000rmb，费用根据持续时间、周作业量、成绩要求有所浮动(持续时间越长约便宜、周作业量越多约贵、成绩要求越高越贵)，报价后价格觉得合适，可以先付一周的款，我们帮你试做，满意后再继续，遇到Fail全额退款。

3、myassignments-help公司所有MATH作业代写服务支持付半款，全款，周付款，周付款一方面方便大家查阅自己的分数，一方面也方便大家资金周转，注意:每周固定周一时先预付下周的定金，不付定金不予继续做。物理代写一次性付清打9.5折。

Math作业代写、数学代写常见问题

myassignments-help擅长领域包含但不是全部: