Posted on Categories:Functional Analysis, 数学代写, 泛函分析

# 数学代写|泛函分析代写Functional Analysis代考|MATH3051 Spectrum and Resolvent

avatest™

avatest泛函分析functional analysis代写，免费提交作业要求， 满意后付款，成绩80\%以下全额退款，安全省心无顾虑。专业硕 博写手团队，所有订单可靠准时，保证 100% 原创。avatest™， 最高质量的matlab作业代写，服务覆盖北美、欧洲、澳洲等 国家。 在代写价格方面，考虑到同学们的经济条件，在保障代写质量的前提下，我们为客户提供最合理的价格。 由于统计Statistics作业种类很多，同时其中的大部分作业在字数上都没有具体要求，因此matlab作业代写的价格不固定。通常在经济学专家查看完作业要求之后会给出报价。作业难度和截止日期对价格也有很大的影响。

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

## 数学代写|泛函分析代写FUNCTIONAL ANALYSIS代考|Spectrum and Resolvent

In Linear Algebra, a complex number $\lambda$ is said to be an eigenvalue of an $(n \times n)$ matrix $A$ with complex coefficients if there exists a nonzero vector $x \in \mathbb{C}^n$ such that $A x=\lambda x$. The number $\lambda$ is an eigenvalue if and only if $\lambda I-A$ fails to be invertible, or equivalently, if and only if $\operatorname{det}(\lambda I-A)=0$. Writing out the determinant we obtain the so-called characteristic polynomial in the variable $\lambda$, which has $n$ zeroes (counting multiplicities) by the main theorem of Algebra. Our first task will be to investigate to what extent these results generalise to bounded operators acting on a Banach space.

Throughout the chapter, $T$ denotes a bounded operator acting on a complex space $X$. We work over the complex scalars; this convention will remain force throughout the rest of this work.

Definition 6.1 (Resolvent and spectrum). The resolvent set of an operator $T \in \mathscr{L}(X)$ is the set $\rho(T)$ consisting of all $\lambda \in \mathbb{C}$ for which the operator $\lambda I-T$ is boundedly invertible, by which we mean that there exists a bounded operator $U$ on $X$ such that
$$(\lambda I-T) U=U(\lambda I-T)=I$$

The spectrum of $T$ is the complement of the resolvent set of $T$ :
$$\sigma(T):=\mathbb{C} \backslash \rho(T) .$$
From now on we shall write $\lambda-T$ instead of $\lambda I-T$. It is customary to write
$$R(\lambda, T):=(\lambda-T)^{-1}$$
for the resolvent operator of $T$ at the point $\lambda \in \rho(T)$. By the open mapping theorem (Theorem 5.8), a complex number $\lambda$ belongs to $\rho(T)$ if and only if $\lambda-T$ is a bijection on $X$.

## 数学代写|泛函分析代写FUNCTIONAL ANALYSIS代考|The holomorphic Functional Calculus

If $f: \mathbb{C} \rightarrow \mathbb{C}$ is an entire function and $T$ is a bounded operator on $X$, we may define a bounded operator $f(T)$ on $X$ as follows. Writing $f$ as a convergent power series about $z=0$,
$$f(z)=\sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n !} z^n$$

we define
$$f(T):=\sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n !} T^n .$$
This series converges absolutely in $\mathscr{L}(X)$ since the same is true for the power series of $f(z)$ for every $z \in \mathbb{C}$. The mapping $f \mapsto f(T)$ is called the entire functional calculus of $T$ and has the following properties, each of which is a consequence of the corresponding properties for scalar-valued entire functions:
(i) if $f(z)=z^n$ with $n \in \mathbb{N}$, then $f(T)=T^n$;
(ii) $f(T) g(T)=(f g)(T)$;
(iii) $g(f(T))=(g \circ f)(T)$.
This calculus may be used to define operators such as $\exp (T), \sin (T), \cos (T)$, and so forth. There is a beautiful way to extend the entire functional calculus to a larger class of holomorphic functions, namely by replacing power series expansions by the Cauchy integral formula
$$f\left(z_0\right)=\frac{1}{2 \pi i} \int_{\Gamma} \frac{f(\lambda)}{\lambda-z_0} \mathrm{~d} \lambda$$

## 数学代写|泛函分析代写FUNCTIONAL ANALYSIS代考|Spectrum and Resolvent

$$(\lambda I-T) U=U(\lambda I-T)=I$$

$$\sigma(T):=\mathbb{C} \backslash \rho(T) .$$

$$R(\lambda, T):=(\lambda-T)^{-1}$$

## 数学代写|泛函分析代写FUNCTIONAL ANALYSIS代考|The holomorphic Functional Calculus

$$f(z)=\sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n !} z^n$$

$$f(T):=\sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n !} T^n$$

(i) 如果 $f(z)=z^n$ 和 $n \in \mathbb{N} ，$ 然后 $f(T)=T^n$ ；
(二) $f(T) g(T)=(f g)(T)$
$\Leftrightarrow g(f(T))=(g \circ f)(T)$.

$$f\left(z_0\right)=\frac{1}{2 \pi i} \int_{\Gamma} \frac{f(\lambda)}{\lambda-z_0} \mathrm{~d} \lambda$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。