arXiv 0810.42402008-10-23math.FA

Estimates for covering numbers in Schauder's theorem about adjoints of compact operators

Michael Cwikel, Eliahu Levy

备注：: 13 pages

英文摘要：

Let T:X --> Y be a bounded linear map between Banach spaces X and Y. Let S:Y' --> X' be its adjoint. Let B(X) and B(Y') be the closed unit balls of X and Y' respectively. We obtain apparently new estimates for the covering numbers of the set S(B(Y')). These are expressed in terms of the covering numbers of T(B(X)), or, more generally, in terms of the covering numbers of a "significant" subset of T(B(X)). The latter more general estimates are best possible. These estimates follow from our new quantitative version of an abstract compactness result which generalizes classical theorems of Arzela-Ascoli and of Schauder. Analogous estimates also hold for the covering numbers of T(B(X)), in terms of the covering numbers of S(B(Y')) or in terms of a suitable "significant" subset of S(B(Y')).

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Schauder's Theorem and s-Numbers

Motivated by the well known theorem of Schauder, we study the relationship between various s-numbers of an operator T and its adjoint T* between Banach spaces.

2022-04-26math.FA

A Schauder and Riesz Basis Criterion for Non-Self-Adjoint Schrödinger Operators with Periodic and Antiperiodic Boundary Conditions

Under the assumption that $V \in L^2([0,π]; dx)$, we derive necessary and sufficient conditions for (non-self-adjoint) Schrödinger operators $-d^2/dx^2+V$ in $L^2([0,π]; dx)$ with periodic and antiperiodic boundary conditions to possess a Riesz basis of root vectors (i.e., eigenvectors and generalized eigenvectors spanning the range of the Riesz projection associated with the corresponding periodic and antiperiodic eigenvalues). We also discuss the case of a Schauder basis for periodic and antiperiodic Schrödinger operators $-d^2/dx^2+V$ in $L^p([0,π]; dx)$, $p \in (1,\infty)$.

2011-06-04math.CAmath.FAmath.SP

A new and simple proof of Schauder's theorem

Schauder's theorem asserts that a bounded linear operator between Banach spaces is compact if ad only if its adjoint is. We give a new proof of this result, which is both short and completely elementary in the sense that it does not depend on anything beyond basic functional analysis, i.e., the Hahn--Banach theorem and some of its consequences; in particular, we avoid the Arzela--Ascoli theorem (and any kind of related diagonal argument).

2011-03-09math.FA

Spectrums of equivalent Schauder operators

Assume that $T_1,T_2$ are equivalent Schauder operators. In this paper, we show that even in this case their Schauder spectrum may be very different in the view of operator theory. In fact, we get that if a self-adjoint Schauder operator $A$ has more than one points in its essential spectrum $σ_e(A)$, then there exists a unitary spread operator $U$ such that the Schauder spectrum $σ_S(UA)$ contains a ring which is depended by the essential spectrum; if there is only one point in $σ_e(A)$ and satisfies some conditions then there exists a unitary spread operator $U$ such that the Schauder spectr…

2012-04-15math.FA

紧度相等：Schauder 定理与 s-数

Equality in Degrees of Compactness: Schauder's Theorem and s-numbers

本文推广 Schauder 定理，研究算子及其伴随算子的 s-数与紧度关系，证明紧算子逼近数相等，并给出非紧算子在自然空间条件及逼近格式下的对应结论。

2023-06-06math.FAmath.SP

On linear operators with s-nuclear adjoints: $0< s \le 1$

If $ s\in (0,1]$ and $ T$ is a linear operator with $ s$-nuclear adjoint from a Banach space $ X$ to a Banach space $ Y$ and if one of the spaces $ X^*$ or $ Y^{***}$ has the approximation property of order $s,$ $AP_s,$ then the operator $ T$ is nuclear. The result is in a sense exact. For example, it is shown that for each $r\in (2/3, 1]$ there exist a Banach space $Z_0$ and a non-nuclear operator $ T: Z_0^{**}\to Z_0$ so that $Z_0^{**}$ has a Schauder basis, $ Z_0^{***}$ has the $AP_s$ for every $s\in (0,r)$ and $T^*$ is $r$-nuclear.

2013-11-10math.FA

更多相关论文收起更多论文

算子空间的一致球覆盖性质的逼近性质与定量估计

Approximation properties and quantitative estimation for uniform ball-covering property of operator spaces

本文通过 Schauder 框架膨胀技术推广了 Godefroy-Kalton 逼近定理，刻画了 λ-UBAP 与 1-UFDD 补子空间嵌入的等价关系，并在此基础上给出算子空间 B(X,Y) 具有一致球覆盖性质的充分条件及定量估计。

2025-07-03math.FA

Iterates of Markov operators and their limits

It is well known that iterates of quasi-compact operators converge towards a spectral projection, whereas the explicit construction of the limiting operator is in general hard to obtain. Here, we show a simple method to explicitly construct this projection operator, provided that the fixed points of the operator and its adjoint are known which is often the case for operators used in approximation theory. We use an approach related to Riesz-Schauder and Fredholm theory to analyze the iterates of operators on general Banach spaces, while our main result remains applicable without specific knowle…

2017-06-02math.FA

Compact bilinear operators on asymmetric normed spaces

The paper is concerned with compact bilinear operators on asymmetric normed spaces. The study of multilinear operators on asymmetric normed spaces was initiated by Latreche and Dahia, Colloq. Math. (2020). We go further in this direction and prove a Schauder type theorem on the compactness of the adjoint of a compact bilinear operator and study the ideal properties of spaces of compact bilinear operators. These extend some results of Ramanujan and Schock, Linear and Multilinear Algebra (1985), and Ruch, ibid. (1989), on compact bilinear operators on Banach spaces. On the space of bilinear form…

2021-11-07math.FA

单边加权移位算子伴随的线性动力学

Linear dynamics of the adjoint of a unilateral weighted shift operator

Indian Institute of Technology Bhubaneswar（印度技术学院布巴内斯瓦尔分校）

本文研究解析函数空间上加权前向移位算子伴随的动力学，给出连续性条件、相似性、本质谱、超循环性、混合性和混沌性，并发现其轨道极限点的零一律不成立。

2026-05-23math.DSmath.FA

Pseudomodes for non-self-adjoint Dirac operators

Depending on the behaviour of the complex-valued electromagnetic potential in the neighbourhood of infinity, pseudomodes of one-dimensional Dirac operators corresponding to large pseudoeigenvalues are constructed. This is a first systematic non-semi-classical approach, which results in substantial progress in achieving optimal conditions and conclusions as well as in covering a wide class of previously inaccessible potentials, including superexponential ones.

2021-12-24math-phmath.APmath.FA

Point Spectrum of Periodic Operators on Universal Covering Trees

For any multi-graph $G$ with edge weights and vertex potential, and its universal covering tree $\mathcal{T}$, we completely characterize the point spectrum of operators $A_{\mathcal{T}}$ on $\mathcal{T}$ arising as pull-backs of local, self-adjoint operators $A_{G}$ on $G$. This builds on work of Aomoto, and includes an alternative proof of the necessary condition for point spectrum he derived in (Aomoto, 1991). Our result gives a finite time algorithm to compute the point spectrum of $A_{\mathcal{T}}$ from the graph $G$, and additionally allows us to show that this point spectrum is containe…

2020-09-29math-phmath.COmath.FA

通过投影与幂零算子进行算子刻画

Operator Characterization via Projectors and Nilpotents

本文研究可数、连续及混合谱算子，引入类似矩阵概念并利用图模型分类，将谱映射定理与von Neumann谱定理推广至更广泛的算子类，为有限维与无限维算子提供统一分类与分解框架。

2024-11-04math.FAmath.OA

Weak Hardy Spaces $WH_L^p({\mathbb R}^n)$ Associated to Operators Satisfying $k$-Davies-Gaffney Estimates

Let $L$ be a one-to-one operator of type $ω$ having a bounded $H_\infty$ functional calculus and satisfying the $k$-Davies-Gaffney estimates with $k\in{\mathbb N}$. In this paper, the authors introduce the weak Hardy space $WH_L^p(\mathbb{R}^n)$ associated to $L$ for $p\in (0,\,1]$ via the non-tangential square function $S_L$ and establish a weak molecular characterization of $WH_L^p(\mathbb{R}^n)$. Typical examples of such operators include the $2k$-order divergence form homogeneous elliptic operator $L_1:=(-1)^k\sum_{|α|=k=|β|}\partial^β(a_{α,β}\partial^α)$, where $\{a_{α,β}\}_{|α|=k=|β|}$ a…

2014-11-30math.CAmath.FA