# 数学代写|复变函数作业代写Complex function代考|AMATH 567

#### Doug I. Jones

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

We are now ready to formulate the Abstract Interpolation Problem AIP based on a data set ${D, \mathcal{T}, \mathbf{T}, E, N}$ described as follows. We are given a linear space $\mathcal{X}$, a positive semidefinite Hermitian form $D$ on $\mathcal{X}$, Hilbert spaces $\mathcal{U}$ and $\mathcal{Y}$, linear operators $\mathfrak{T}, \mathbf{T}=\left(T_{1}, \ldots, T_{d}\right)$ on $\mathcal{X}$, and linear operators $N: \mathcal{X} \rightarrow \mathcal{U}$ and $E: \mathcal{X} \rightarrow$ $\mathcal{Y}$. In addition we assume that
$$D(\mathfrak{T} x, \mathfrak{T} x)+|N x|_{\mathcal{L}}^{2}=\sum_{j=1}^{d} D\left(T_{j} x, T_{j} x\right)+|E x|_{\mathcal{Y}}^{2} \quad \text { for every } x \in \mathcal{X}$$
Definition 7.1 Suppose that we are given the data set ${D, T, \mathbf{T}, E, N}$ for an AIP as in (7.1). We say that the pair $(F, S)$ is a solution of the AIP if $S \in \mathcal{S}{\mathrm{nc}, d}(\mathcal{U}, \mathcal{Y})$ and $F$ is a linear mapping from $\mathcal{X}$ into $\mathcal{H}\left(K{S}\right)$ such that
\begin{aligned} &|F x|_{\mathcal{H}\left(K_{S}\right)}^{2} \leq D(x, x) \quad \text { for all } \quad x \in \mathcal{X} \ &(F \mathfrak{T} x)(z)-\sum_{j=1}^{d}\left(F T_{j} x\right)(z) z_{j}=(E-S(z) N) x \end{aligned}
Denote by $\mathcal{X}{0}$ the Hilbert space equal to the completion of the space of equivalence classes of elements of $\mathcal{X}$ (where the zero equivalence class consists of elements $x$ with $D(x, x)=0$ ) in the $D$-inner product. Note that if $(S, F)$ solves AIP, then condition (7.2) implies that $F$ has a factorization $F{0} \circ \pi$ where $\pi$ is the canonical projection operator $\pi: \mathcal{X} \rightarrow \mathcal{X}{0}$ and where $F{0}: \mathcal{X}{0} \rightarrow \mathcal{H}\left(K{S}\right)$ has $\left|F_{0}\right| \leq 1$. We abuse notation and denote also by $T$ and $T_{k}$ the operators $T$ and $T_{k}$ followed by the canonical projection into the equivalence class in $\mathcal{X}{0} ;$ so $\mathfrak{T}, T{k}: \mathcal{X} \rightarrow \mathcal{X}{0}$. Let for short $$T=\left[\begin{array}{c} T{1} \ \vdots \ T_{d} \end{array}\right]: \mathcal{X} \rightarrow \mathcal{X}{0}^{d}, \quad Z(z)=\left[z{1} I_{\mathcal{X}{0}} \cdots z{d} I_{\mathcal{X}_{0}}\right]$$

## 数学代写|复变函数作业代写Complex function代考|Regular Extensions and Defect Functions

Let $\mathfrak{G}$ and $\mathfrak{F}$ be Hilbert spaces (all Hilbert spaces considered in this paper are assumed to be complex and separable). By $[\mathfrak{G}, \mathfrak{F}]$ we denote the Banach space of bounded linear operators defined on $\mathfrak{G}$ and taking values in $\mathfrak{F}$. If $\mathfrak{F}=\mathfrak{G}$, we use the notation $[\mathfrak{G}]:=[\mathfrak{G}, \mathfrak{G}]$

Let $\mathbb{D}:={\zeta \in \mathbb{C}:|\zeta|<1}$ and $\mathbb{T}:={z \in \mathbb{C}:|z|=1}$. By $L^{\infty}[\mathfrak{G}, \mathfrak{F}]$ we denote the Banach space of measurable (indifferently in what sense, weak or strong, in view of the separability of the spaces $\mathfrak{G}$ and $\mathfrak{F})[\mathfrak{G}, \mathfrak{F}$-valued functions $\theta(z), z \in \mathbb{T}$, such that
$$|\theta|_{L^{\infty}[\mathfrak{G}, \mathfrak{F}]}:=\operatorname{ess} \sup {z}|\theta(z)|{[\mathfrak{G}, \mathfrak{F}]}<\infty \text {. }$$
Functions belonging to the closed unit ball
$$C M[\mathfrak{G}, \mathfrak{F}]:=\left{\theta(z):|\theta|_{L^{\infty}[\mathfrak{G}, \mathfrak{F}]} \leq 1\right}$$
of the space $L^{\infty}[\mathfrak{G}, \mathfrak{F}]$ are called contractive measurable $[\mathfrak{G}, \mathfrak{F}]$ – valued functions.
If $H_{+}^{\infty}[\mathfrak{G}, \mathfrak{F}]$ is the Hardy space of bounded holomorphic $[\mathfrak{G}, \mathfrak{F}]$-valued functions $\theta(\zeta), \zeta \in \mathbb{D}$, that is, such that
$$|\theta|_{H_{+}^{\infty}[\mathfrak{G}, \mathfrak{F}]}:=\sup {\zeta}|\theta(\zeta)|{[\mathfrak{G}, \mathfrak{F}]}<\infty,$$
then by $L_{+}^{\infty}[\mathfrak{G}, \mathfrak{F}]$ we denote the subspace of $L^{\infty}[\mathfrak{G}, \mathfrak{F}]$ consisting of strong boundary value functions $\theta(z)$ for $\theta(\zeta) \in H_{+}^{\infty}[\mathfrak{G}, \mathfrak{F}]$. Moreover, the equality
$$|\theta(z)|_{L_{+}^{\infty}[\mathfrak{G}, \mathfrak{F}]}=|\theta(\zeta)|_{H_{+}^{\infty}[\mathfrak{G}, \mathfrak{F}]}$$
makes it possible to identify the spaces $H_{+}^{\infty}[\mathfrak{G}, \mathfrak{F}]$ and $L_{+}^{\infty}[\mathfrak{G}, \mathfrak{F}]$ up to the obvious isomorphism. Functions belonging to the closed unit ball
$$S[\mathfrak{G}, \mathfrak{F}]:=\left{\theta(\zeta):|\theta|_{H_{+}^{\infty}[\mathfrak{G}, \mathfrak{F}]} \leq 1\right}$$
of the space $H_{+}^{\infty}[\mathfrak{G}, \mathfrak{F}]$ are usually called $S c h u r[\mathfrak{G}, \mathfrak{F}]$-valued functions.

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

$\mathfrak{T}, \mathbf{T}=\left(T_{1}, \ldots, T_{d}\right)$ 上 $\mathcal{X}$, 和线性算子 $N: \mathcal{X} \rightarrow \mathcal{U}$ 和 $E: \mathcal{X} \rightarrow \mathcal{Y}$. 此外，我们假设 $D(\mathfrak{T} x, \mathfrak{T} x)+|N x|{\mathcal{L}}^{2}=\sum{j=1}^{d} D\left(T_{j} x, T_{j} x\right)+|E x|{\mathcal{Y}}^{2} \quad$ for every $x \in \mathcal{X}$ 定义 $7.1$ 假设给定数据集 $D, T, \mathbf{T}, E, N$ 对于 $(7.1)$ 中的 AIP。我们说这对 $(F, S)$ 是 AIP 的解决方案，如果 $S \in \mathcal{S n c}, d(\mathcal{U}, \mathcal{Y})$ 和 $F$ 是一个线性映射 $\mathcal{X}$ 进入 $\mathcal{H}(K S)$ 这样 $|F x|{\mathcal{H}\left(K_{S}\right)}^{2} \leq D(x, x) \quad$ for all $\quad x \in \mathcal{X} \quad(F \mathfrak{T} x)(z)-\sum_{j=1}^{d}\left(F T_{j} x\right)(z) z_{j}=$

$F 0: \mathcal{X} 0 \rightarrow \mathcal{H}(K S)$ 有 $\left|F_{0}\right| \leq 1$. 我们滥用符昊并且还表示为 $T$ 和 $T_{k}$ 运营商 $T$ 和 $T_{k}$ 然 后是到等价类的规范投影 $\mathcal{X} 0 ;$ 所以吉, $T k: \mathcal{X} \rightarrow \mathcal{X} 0$. 让或们简称
$$T=\left[T 1: T_{d}\right]: \mathcal{X} \rightarrow \mathcal{X} 0^{d}, \quad Z(z)=\left[z 1 I_{\mathcal{X}{0}} \cdots z d I{\mathcal{X}_{0}}\right]$$

## 数学代写|复变函数作业代写Complex function代考|Regular Extensions and Defect Functions

$$|\theta|{L^{\infty}[\mathfrak{G}, \mathfrak{₹}]}:=\operatorname{ess} \sup z|\theta(z)|[\mathfrak{G}, \mathfrak{F}]<\infty$$ 属于封闭单元球的功能 C M[/mathfrak{G}, \mathfrak{F}]:=|left{theta(z): | theta $\left.\right|{-}\left{L^{\wedge}{\backslash i n f t y}[\backslash m a t h f r a k{G}\right.$, Imathfra

$$|\theta|{H{+}^{\infty}[\mathfrak{G}, \mathfrak{F}]}:=\sup \zeta|\theta(\zeta)|[\mathfrak{G}, \mathfrak{F}]<\infty$$

$$|\theta(z)|{L{+}^{\infty}[\mathfrak{G}, \mathfrak{F}]}=|\theta(\zeta)|{H{+}^{\infty}[\mathfrak{G}, \mathfrak{F}]}$$

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