Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 0707.3801

$N_{\p}$-type quotient modules on the torus

Keiji Izuchi·Rongwei Yang
View original source ↗Read source document ↗Export metadata ↓

Abstract

Structure of the quotient modules in $\hh$ is very complicated. A good understanding of some special examples will shed light on the general picture. This paper studies the so-call $N_{\p}$-type quotient modules, namely, quotient modules of the form $\hh\ominus [z-\p]$, where $\p (w)$ is a function in the classical Hardy space $H^2(\G)$ and $[z-\p]$ is the submodule generated by $z-\p (w)$. This type of quotient modules serve as good examples in many studies. A notable feature of the $N_{\p}$-type quotient module is its close connections with some classical single variable operator theories.

Explore related subjects

math.FA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Keiji Izuchi, Rongwei Yang. 2007-07-25. $N_{\p}$-type quotient modules on the torus. https://arxiv.org/abs/0707.3801

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The best Ulam constant of the n-th order linear differential operator with constant coefficients

Let $X$ be a Banach space over the field $\mathbb{K} \in \{\mathbb{R}, \mathbb{C}\}$, and let $L:\mathcal{C}^{n}(\mathbb{R},X)\to\mathcal{C}(\mathbb{R},X)$ be the $n$-th order linear differential operator defined by $Ly := y^{(n)} + a_1 y^{(n-1)} + \cdots + a_n y$, where $a_1,\ldots,a_n \in \mathbb{K}$ are fixed scalars and $n$ is a fixed positive integer. It is well known that $L$ is Ulam stable if and only if its characteristic equation has no roots on the imaginary axis. In this paper, we determine the best Ulam constant of $L$ under this condition, for an arbitrary order $n$ and without any restriction on the multiplicities of the roots of the characteristic equation. This improves and completes several known Ulam stability results for this operator and provides the best estimate possible for the error between approximate and exact solutions of the equation $Ly=f$, where $f\in\mathcal{C}(\mathbb{R},X)$. Moreover, as an auxiliary result, we give an explicit formula for the general solution of this equation.

math.FA↗
arXiv2026

The Sharp Scale Inequality for the Schatten Trace Class

In this paper, we establish the metric cotype $2$ inequality for the Schatten trace class $S_{1}$ at the optimal order of the scaling parameter. More precisely, for every integer $n\geq 1$, every even integer $m\geq 4$, and every mapping $f:\mathbb{Z}_{m}^{n}\to S_{1}$, the following holds \begin{align*} &\left(\sum_{j=1}^{n}\mathbb{E}_{x}\left\|f\left(x+\frac{m}{2}e_{j}\right)-f(x)\right\|_{1}^{2}\right)^{1/2}\\ \leq&\left(3\sqrt{70}\,m+6\sqrt{n}\right)\left(\mathbb{E}_{x,\varepsilon}\left\|f(x+\varepsilon)-f(x)\right\|_{1}^{2}\right)^{1/2}, \end{align*} where $x$ and $\varepsilon$ are independently uniform on $\mathbb{Z}_{m}^{n}$ and $\{-1,0,1\}^{n}$, respectively. The order $\sqrt{n}$ is necessary for any fixed metric cotype constant. This settles the optimal scaling problem for the Schatten trace class posed by Mendel and Naor \cite[Section 8]{MN} (see also \cite[Question 3]{GMN}).

math.FA↗
arXiv2026

Non-Archimedean Heisenberg-Robertson-Schrodinger and Non-Archimedean Maccone-Pati Uncertainty Principles

Let $\mathcal{X}$ be a non-Archimedean Hilbert space over a conjugated non-Archimedean valued field $\mathbb{K}$ with $\operatorname{Char}(\mathbb{K})\neq 2$. Let $A: \mathcal{D}(A)\subseteq \mathcal{X}\to \mathcal{X}$ and $B: \mathcal{D}(B)\subseteq \mathcal{X}\to \mathcal{X}$ be possibly unbounded self-adjoint linear operators. For $x \in \mathcal{D}(A)$ with $\langle x, x \rangle =1$, define $ Δ_x(A):= \|Ax- \langle Ax, x \rangle x \|.$ Then for all $x \in \mathcal{D}(AB)\cap \mathcal{D}(BA)$ with $\langle x, x \rangle =1$, we show that \begin{align*} (1) \quad Δ_x(A)+Δ_x(B)\geq \max\{Δ_x(A), Δ_x(B)\}\geq \frac{\sqrt{\bigg|\big\langle [A,B]x, x \big\rangle ^2+\big(\langle \{A,B\}x, x \rangle -2\langle Ax, x \rangle\langle Bx, x \rangle\big)^2\bigg|}}{\sqrt{|2|}} \end{align*} and \begin{align*} (2) \quad Δ_x(A)+Δ_x(B)\geq \max\{Δ_x(A), Δ_x(B)\} \geq |\langle (A+B)x, y \rangle |, \quad \forall y \in \mathcal{X} \text{ satisfying } \|y\|\leq 1, \langle x, y \rangle =0. \end{align*} Inequality (1) may be regarded as a non-Archimedean analogue of the Heisenberg-Robertson-Schrodinger uncertainty principle, while Inequality (2) serves as a non-Archimedean analogue of the Maccone-Pati uncertainty principle.

math.FA↗