### 数学代写|复变函数作业代写Complex function代考|Math 417

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## 数学代写|复变函数作业代写Complex function代考|The Influence of V. E. Katsnelson and D. Z. Arov on the Direction of Our Research Group

While working on generalized matricial Nehari problems (see $[21]$ ), Bernd and I made first contact with the works of V. E. Katsnelson. The problem is stated as follows:
GENERALIZED MATRICIAL NEHARI PROBLEM: Let $p, q \in \mathbb{N}$. Further, let $F_{11}$ and $F_{22}$ be a non-negative Hermitian $p \times p$ and a $q \times q$ measure, respectively, on the Borelian $\sigma$-Algebra $\mathfrak{B}{\mathbb{T}}$ on $\mathbb{T}:={z \in \mathbb{C}:|z|=1}$ and let $\left(\beta{k}\right){k=0}^{\infty}$ be a sequence of complex $p \times q$ matrices. Describe the set $\mathcal{F}\left(F{11}, F_{22},\left(\beta_{k}\right){k=0}^{\infty}\right)$ of all $\sigma$-additive mappings $F{12}$ from $\mathfrak{B}{\mathrm{T}}$ into the set of all complex $p \times q$ matrices fulfilling the conditions $$\int{\mathbb{T}} z^{-k} F_{12}(\mathrm{~d} z)=\beta_{k}, \quad k=0,1,2, \ldots$$
and for which
$$\left(\begin{array}{ll} F_{11} & F_{12} \ F_{12}^{} & F_{22} \end{array}\right)$$ is a non-negative Hermitian $(p+q) \times(p+q)$ measure on $\mathfrak{B}{\mathbb{T}}$. In particular, state necessary and sufficient conditions such that the set $\mathcal{F}\left(F{11}, F_{22},\left(\beta_{k}\right){k=0}^{\infty}\right)$ is non-empty. The problem stated above leads one to studying kernels on $\mathbb{N}{0} \times \mathbb{N}{0}$ of so-called mixed Toeplitz-Hankel type. To see this, for all $k \in \mathbb{Z}$, set $$\alpha{k}:=\int_{\mathbb{T}} z^{-k} F_{11}(\mathrm{~d} z) \quad \text { and } \quad \delta_{k}:=\int_{\mathbb{T}} z^{-k} F_{22}(\mathrm{~d} z)$$
and, for all $(m, n) \in \mathbb{N}{0} \times \mathbb{N}{0}$, define
$$K(m, n):=\left(\begin{array}{cc} \alpha_{m-n} & \beta_{m+n} \ \beta_{m+n}^{} & \delta_{n-m} \end{array}\right)$$
The kernel $K$ being non-negative definite turns out to be necessary and sufficient for the set $\mathcal{F}\left(F_{11}, F_{22},\left(\beta_{k}\right)_{k=0}^{\infty}\right)$ to be non-empty.

The just defined kernel $K$ is also important because of the following observation.
GENERALIZED HERGLOTZ-BOCHNER THEOREM: Let $p, q \in \mathbb{N}$ and let $\left(\alpha_{k}\right){k=0}^{\infty},\left(\beta{k}\right){k=0}^{\infty}$, and $\left(\delta{k}\right){k=0}^{\infty}$ be sequences belonging to $\mathbb{C}^{p \times p}, \mathbb{C}^{p \times q}$, and $\mathbb{C}^{q \times q}$, respectively. Then there exists a non-negative Hermitian $(p+q) \times(p+q)$ Borelian measure on T such that for all $m, n \in{0,1,2, \ldots}$ the equation $$K(m, n)=\int{\mathbb{T}}\left[\operatorname{diag}\left(z^{-m} I_{p}, z^{m} I_{q}\right)\right] F(\mathrm{~d} z)\left[\operatorname{diag}\left(z^{-n} I_{p}, z^{n} I_{q}\right)\right]^{*}$$
is satisfied if and only if $K$ is non-negative definite.

## 数学代写|复变函数作业代写Complex function代考|A Preamble

As a mathematician, Victor Katsnelson was raised within a fine school of function theory and functional analysis, which was blossoming in Kharkov starting the second half of 1930s. He studied in the Kharkov State University in 1960-1965. Among his teachers were Naum Akhiezer, Boris Levin, Vladimir Marchenko. That time he became acquainted with Vladimir Matsaev whom Victor often mentions as one of his teachers. In 1965 Katsnelson graduated with the master degree, Boris Levin supervised his master thesis. Since then and till 1990, he teaches at the Department of Mathematics and Mechanics of the Kharkov State University. In 1967 he defends the PhD Thesis “Convergence and Summability of Series in Root Vectors of Some Classes of Non-Selfadjoint Operators” also written under Boris Levin guidance. Until he left Kharkov in the early 1990s, Katsnelson remained an active participant of the Kharkov function theory seminar run on Thursdays by Boris Levin and Iossif Ostrovskii. His talks, remarks and questions were always interesting and witty.

Already in the 1960s Victor established himself among the colleagues as one of the finest Kharkov mathematicians of his generation, if not the finest one. Nevertheless, he was not appointed as a professor and was never allowed to travel abroad.

Most of Katsnelson’s work pertain to the spectral theory of functions and operators. I will touch only a handful of his results, mostly published in 1965-1970,that is, at the very beginning of his mathematical career. A big portion of his works written in Kharkov appeared in the local journal “Function Theory, Functional Analysis and Their Applications” and were never translated in English. Today, this journal is available at http://dspace.univer.kharkov.ua/handle/123456789/43.

In this occasion, let me mention two wonderful books carefully written by Katsnelson $[18,19]$. They exist only as manuscripts, and curiously, both have “Part I” in their titles, though, as far as I know, no continuations appeared. In both books mathematics interlaces with interesting historical comments. Last but not least, let me also mention an extensive survey of Issai Schur’s works in analysis written jointly by Dym and Katsnelson [7].

## 数学代写|复变函数作业代写Complex function代考|The Influence of V. E. Katsnelson and D. Z. Arov on the Direction of Our Research Group

GENERALIZED MATRICIAL NEHARI PROBLEM: Let $p, q \in \mathbb{N}$. 此外，让 $F_{11}$ 和 $F_{22}$ 是一个非负厄米特 $p \times p$ 和一个 $q \times q$ 分别测量 Borelian $\sigma$-代数 $\mathfrak{B T}$ 上T $:=z \in \mathbb{C}:|z|=1$ 然后让 $(\beta k) k=0^{\infty}$ 是一个昆杂的序列 $p \times q$ 矩阵。描述集合 $\mathcal{F}\left(F 11, F_{22},\left(\beta_{k}\right) k=0^{\infty}\right)$ 其中 $\sigma$ – 加法映射 $F 12$ 从 $\mathfrak{B}$ T 进入所有复数的集合 $p \times q$ 满足条件的矩阵
$$\int \mathbb{T} z^{-k} F_{12}(\mathrm{~d} z)=\beta_{k}, \quad k=0,1,2, \ldots$$

$$\left(\begin{array}{llll} F_{11} & F_{12} & F_{12} & F_{22} \end{array}\right)$$

$\mathcal{F}\left(F 11, F_{22},\left(\beta_{k}\right) k=0^{\infty}\right)$ 是非空的。上述问题导致研究内核 $\mathbb{N} 0 \times \mathbb{N} 0$ 所调的混合 Toeplitz-Hankel 类型。看到这 个，给大家 $k \in \mathbb{Z}$ ，放
$$\alpha k:=\int_{\mathbb{T}} z^{-k} F_{11}(\mathrm{~d} z) \quad \text { and } \quad \delta_{k}:=\int_{\mathbb{T}} z^{-k} F_{22}(\mathrm{~d} z)$$

$$K(m, n)=\int \mathbb{T}\left[\operatorname{diag}\left(z^{-m} I_{p}, z^{m} I_{q}\right)\right] F(\mathrm{~d} z)\left[\operatorname{diag}\left(z^{-n} I_{p}, z^{n} I_{q}\right)\right]^{*}$$

## 数学代写|复变函数作业代写Complex function代考|A Preamble

Katsnelson 的大部分工作都与函数和算子的谱理论有关。我将只触及他的一小部分结果，大部分发表于 1965-1970 年，也就是他数学生涯的初期。他用哈尔科夫写的大部分作品出现在当地期刊《泛函理论、泛函分析及其应用》上，从未翻译成英文。今天，该期刊可在 http://dspace.univer.kharkov.ua/handle/123456789/43 获得。

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