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## 数学代写|复变函数作业代写Complex function代考|Operator Argument Interpolation Problem: Solution

In this section we present a solution of the Operator Argument Interpolation Problem, including the parametrization of the set of all solutions for the case where the operator $P(4.5)$ is invertible. The setting here is more general than the case handled by the Commutant Lifting Theorem discussed in the previous section in that we no longer insist that $\mathbf{T}$ be strongly stable. As a first step we present several useful reformulations of the problem. The main tool for this analysis is the following well-known Hilbert space result. In case the block $A$ is invertible, the result can be seen as a consequence of a standard Schur-complement computation; the general result then follows by replacing $A$ with $A+\epsilon I$ and then letting $\epsilon>0$ tend to zero.
Proposition 5.1 A Hilbert space operator
$$\left[\begin{array}{ll} P & B^{*} \ B & A \end{array}\right]:\left[\begin{array}{l} \mathcal{X} \ \mathcal{H} \end{array}\right] \rightarrow\left[\begin{array}{l} \mathcal{X} \ \mathcal{H} \end{array}\right]$$
is positive semidefinite if and only if $A$ is positive semidefinite and for every $x \in \mathcal{X}$, there exists a vector $h_{x} \in \mathcal{H} \ominus$ Ker $A$ such that
$$A^{\frac{1}{2}} h_{x}=B x \quad \text { and } \quad\left|h_{x}\right|_{\mathcal{H}} \leq\left|P^{\frac{1}{2}} x\right|_{\mathcal{X}}$$
Theorem $5.2$ Given the data set ${\mathbf{T}, E, N}$ such that the pairs $(E, \mathbf{T})$ and $(N, \mathbf{T})$ are output stable, let $P: \mathcal{X} \rightarrow \mathcal{X}$ be defined as in (4.5). Given $S \in \mathcal{L}(\mathcal{U}, \mathcal{Y})\langle\langle z\rangle\rangle$, let $F^{S}: \mathcal{X} \rightarrow H_{\mathcal{Y}}^{2}\left(\mathbb{F}{d}^{+}\right)$be the linear map defined by $$F^{S}: x \mapsto\left(\mathcal{O}{E, \mathbf{T}}-M_{S} \mathcal{O}_{N, \mathbf{T}}\right) x$$

## 数学代写|复变函数作业代写Complex function代考|The Analytic Abstract Interpolation Problem

Besides the left-tangential evaluation calculus (4.1), there is another way to evaluate a formal power series $f(z)=\sum_{\alpha \in \mathbb{F}{d}^{+}} f{\alpha} z^{\alpha} \in \mathcal{Y}\langle\langle z\rangle\rangle$ on a $d$-tuple $\mathbf{Z}=$ $\left(Z_{1}, \ldots, Z_{d}\right) \in \mathcal{L}(\mathcal{X})^{d}$, namely,
$$f(\mathbf{Z})-\sum_{\alpha \in \mathbb{F}{d}^{+}} f{\alpha} \otimes \mathbf{Z}^{\alpha}-\lim {N \rightarrow \infty} \sum{\alpha \in \mathbb{F}{d}^{+}:|\alpha| \leq N} f{\alpha} \otimes \mathbf{Z}^{\alpha} \in \mathcal{Y} \otimes \mathcal{L}(\mathcal{X}),$$

provided the latter limit exists at least in the weak sense. The existence of the weak limit clearly depends on $f$ and $\mathbf{Z}$. Let us denote by $\mathfrak{B}{d}$ the set of all Hilbert space strict row contractions $$\mathfrak{B}{d}=\left{\mathbf{Z}=\left(Z_{1}, \ldots, Z_{d}\right) \in \mathcal{L}(\mathcal{X})^{d}: \sum_{j=1}^{d} Z_{j} Z_{j}^{*} \prec I_{\mathcal{X}}\right}$$
and let us introduce the space $\mathcal{H} \mathcal{Y}\left(\mathfrak{B}{d}\right)$ of formal power series $f \in \mathcal{Y}\langle\langle z\rangle\rangle$ such that the weak limit (6.1) exists for any $d$-tuple $\mathbf{Z} \in \mathfrak{B}{d}$. Observe that by Cauchy inequality, $H_{\mathcal{Y}}^{2}\left(\mathbb{F}{d}^{+}\right) \subset \mathcal{H}{\mathcal{Y}}\left(\mathfrak{B}{d}\right)$. In particular, for an output-stable pair $(C, \mathbf{T})$ and any $x \in \mathcal{X}$, the power series $\mathcal{O}{C, \mathbf{T}} x$ belongs to $\mathcal{H}{\mathcal{Y}}\left(\mathfrak{B}{d}\right)$. Various spaces of such “free holomorphic functions” have been studied systematically in a series of papers by Popescu $[31,32]$; when one restricts the Hilbert space $\mathcal{X}$ to be finitedimensional $\mathcal{X}=\mathbb{C}^{n}$ and then defines $\mathfrak{B}_{d}$ to be the disjoint union of these unit balls of row contraction matrices over $n=1,2,3, \ldots$, one comes into the setting of “free noncommutative functions” which is an area of active current interest (see $[12,13,34])$

The very formulation of the problem $\mathbf{O A P}(\mathbf{T}, E, N)$ appears to require that the operators $\mathcal{O}{E, \mathbf{T}}$ and $\mathcal{O}{N, \mathbf{T}}$ be bounded operators from $\mathcal{X}$ into $H_{\mathcal{Y}}^{2},\left(\mathbb{F}{d}^{+}\right)$and $H{\mathcal{U}}^{2}\left(\mathbb{F}{d}^{+}\right)$ respectively. However, upon close inspection, one can see that conditions (2)-(5) in Theorem $5.2$ make sense and moreover, conditions (2), (3), (4) are mutually equivalent if we only assume that (a) $\mathbf{T}=\left(T{1}, \ldots, T_{d}\right) \in \mathcal{L}(\mathcal{X})^{d}, E: \mathcal{X} \rightarrow \mathcal{Y}$ and $N: \mathcal{X} \rightarrow \mathcal{U}$ are such that
(b) $P$ is a positive semidefinite solution of the Stein equation ( $3.30)$.

## 数学代写|复变函数作业代写Complex function代考|Operator Argument Interpolation Problem: Solution

$$A^{\frac{1}{2}} h_{x}=B x \quad \text { and } \quad\left|h_{x}\right|{\mathcal{H}} \leq\left|P^{\frac{1}{2}} x\right|{\mathcal{X}}$$

$$F^{S}: x \mapsto\left(\mathcal{O} E, \mathbf{T}-M_{S} \mathcal{O}_{N, \mathbf{T}}\right) x$$

## 数学代写|复变函数作业代写Complex function代考|The Analytic Abstract Interpolation Problem

$$f(\mathbf{Z})-\sum_{\alpha \in \mathbb{F} d^{+}} f \alpha \otimes \mathbf{Z}^{\alpha}-\lim N \rightarrow \infty \sum \alpha \in \mathbb{F} d^{+}:|\alpha| \leq N f \alpha \otimes \mathbf{Z}^{\alpha} \in \mathcal{Y} \otimes \mathcal{L}(\mathcal{X}),$$

$\backslash 1$ eft 的分隔符缺失或无法识别

(b) $P$ 是 Stein 方程的半正定解 (3.30).

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## 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。

assignmentutor™您的专属作业导师
assignmentutor™您的专属作业导师