# 电子工程代写|信号处理与线性系统作业代写Signal Processing and Linear Systems代考|EE483

## 电子工程代写|信号处理与线性系统作业代写Signal Processing and Linear Systems代考|Spectral theory of the graph Laplacian in the energy space

In this section, we consider the graph Laplace operator $\Delta$ acting in the energy space $\mathcal{H}=\mathcal{H}_{E}$. We will also discuss the properties of this operator $\Delta$.

Our approach is based on the notion of symmetric pairs of operators. We briefly describe this approach. For more details regarding the theory of unbounded operators, readers may consult the following items [DS88, JT17b] and the papers cited there.

Let $\mathcal{H}{1}$ and $\mathcal{H}{2}$ be Hilbert spaces, and let $\mathcal{D}{1} \subset \mathcal{H}{1}$ and $\mathcal{D}{2} \subset \mathcal{H}{2}$ be dense subspaces. Suppose that two linear operators
$$J: \mathcal{D}{1} \rightarrow \mathcal{H}{2}, \quad K: \mathcal{D}{2} \rightarrow \mathcal{H}{1}$$
are defined on these dense subspaces. The pair $(J, K)$ is called a symmetric pair if
$$\langle J \varphi, \psi\rangle_{\mathcal{H}{2}}=\langle\varphi, K \psi\rangle{\mathcal{H}{1},} \varphi \in \mathcal{D}{1}, \psi \in \mathcal{D}_{2} .$$
The following statement is a well-known result in the theory of unbounded operators.

Lemma 8.1.
(1) Suppose $(J, K)$ be a symmetric pair satisfying (8.1) and (8.2). Then the operators $J$ and $K$ are closable and $J \subset K^{}, K \subset J^{}$. Without loss of generality, one can assume that $J=\bar{J}, K=\bar{K}$.
(2) $J^{} J$ is a self-adjoint densely defined operator in $\mathcal{H}{1}$, and $K^{} K$ is a self-adjoint densely defined operator in $\mathcal{H}{2}$.

Now we apply the above statement to the case of Hilbert spaces $L^{2}(\mu)$ and $\mathcal{H}{E}$. To distinguish the graph Laplace operators acting in $L^{2}(\mu)$ and $\mathcal{H}{E}$, we will use the notation $\Delta_{2}$ and $\Delta_{\mathcal{H}}$, respectively.

As was proved in Theorems $7.3$ and $7.7$, the operator $\Delta_{2}$ is positive definite and essentially self-adjoint; therefore, by the spectral theorem, there exists a projection-valued measure $Q(d t)$ such that
$$\Delta_{2}=\int_{0}^{\infty} t d Q(t)$$
or, for any $\varphi \in L^{2}(\mu)$,
$$\left\langle\varphi, \Delta_{2} \varphi\right\rangle_{L^{2}(\mu)}=\int_{0}^{\infty} t|Q(d t) \varphi|_{L^{2}(\mu)}^{2}$$
(we used here the fact that $Q(d t)$ is a projection).

## 电子工程代写|信号处理与线性系统作业代写Signal Processing and Linear Systems代考|Combinatorial description of the c-free convolution

Let now the pair of probability measures on $\mathbb{R}$ with all moments $\left(\mu_{1}, \nu_{1}\right),\left(\mu_{2}, \nu_{2}\right)$ be given. We will identify $\mu_{i}, \nu_{i}$ with states on polynomial algebras $\mathbb{C}\langle\mathbb{X}\rangle,(i=1,2)$ by the formulas (1). Let us consider the $c$-free product $\varphi=\left(\mu_{1}, \nu_{1}\right) *{c}\left(\mu{2}, \nu_{2}\right)$ on $\mathbb{C}\left\langle\mathbb{X}{1}\right\rangle * \mathbb{C}\left\langle\mathbb{X}{2}\right\rangle=\mathbb{C}\left\langle\mathbb{X}{1}, \mathbb{X}{2}\right\rangle$ – later is the algebra of polynomials is the noncommutative variables $\mathbb{X}{1}$ and $\mathbb{X}{2}$.
The $c$-free convolution
$$\mu=\left(\mu_{1}, \nu_{1}\right) \boxplus\left(\mu_{2}, \nu_{2}\right) \in \mathcal{M}$$
is then givenn as the distribution of $\mathbb{X}:=\mathbb{X}{1}+\mathbb{X}{2}$, i.é.,
$$\int_{\mathbb{R}} t^{n} d \mu(t)-\mu\left(\mathbb{X}^{n}\right)-\varphi\left(\left(\mathbb{X}{1}+\mathbb{X}{2}\right)^{n}\right), \quad n \geq 0 .$$
Note, that if $\mu_{i}=\nu_{i}$, then this reduces to the free convolution of probability measures done by D. Voiculescu [34].

As in the free case there is a nice combinatorial description of $c$-free convolution using the notion of non-crossing partitions $-N C(n)$.

Definition 3.1. Let $\pi=\left{V_{1}, V_{2}, \ldots, V_{p}\right}$ be a partition of the linear outer set ${1,2, \ldots, n}$, i.e., $V_{i} \neq \emptyset$, disjoint $\bigcup V_{j}={1,2, \ldots, n}$. Then $\pi$ is called noncrossing partition if $a, c \in V_{i}$ and $b, d \in V_{j}$, and $a<b<c<d$ implies $i=j$.
The sets $V_{i} \in \pi$ are called blocks.
A block $V_{i}$ of a non-crossing partition $\pi=\left{V_{1}, V_{2}, \ldots, V_{p}\right}$ is called inner block if there exist a $V_{j} \in \pi$ and $a, b, \in V_{j}$ such that $a<v<b$ for at least one (and hence for all) $v \in V_{i}$. A block $V_{i} \in \pi$ which is not inner is called outer.

We will denote by $N C(n)$ the set of all non-crossing partitions of ${1,2, \ldots, n}$, and by $\pi=\left{V_{1}, V_{2}, \ldots, V_{n}\right} \in N C_{2}(2 n)$ we denote those non-crossing partitions where each block $V_{i} \in \pi$ consists of exactly two elements.

## 电子工程代写|信号处理与线性系统作业代写Signal Processing and Linear Systems代考|Spectral theory of the graph Laplacian in the energy space

$$J: \mathcal{D} 1 \rightarrow \mathcal{H} 2, \quad K: \mathcal{D} 2 \rightarrow \mathcal{H} 1$$

$$\langle J \varphi, \psi\rangle_{\mathcal{H} 2}=\langle\varphi, K \psi\rangle \mathcal{H} 1, \varphi \in \mathcal{D} 1, \psi \in \mathcal{D}_{2} .$$

## 电子工程代写|信号处理与线性系统作业代写Signal Processing and Linear Systems代考|Combinatorial description of the c-free convolution

$$\mu=\left(\mu_{1}, \nu_{1}\right) \boxplus\left(\mu_{2}, \nu_{2}\right) \in \mathcal{M}$$

$1,2, \ldots, n$ ，那是， $V_{i} \neq \emptyset$ ，不相交 $\bigcup V_{j}=1,2, \ldots, n$. 然后 $\pi$ 称为非交叉 分区，如果 $a, c \in V_{i}$ 和 $b, d \in V_{j}$ ，和 $a<b<c<d$ 暗示 $i=j$.

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: