scholarly journals The ideal of Lipschitz classical p-compact operators and its injective hull

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Toufik Tiaiba ◽  
Dahmane Achour
Keyword(s):  
Banach Space ◽  
Compact Operator ◽  
Metric Space ◽  
Metric Spaces ◽  
Compact Operators ◽  
Operator Ideal ◽  
Linear Case ◽  
Injective Hull ◽  
Nuclear Operators ◽  
The Ideal

Abstract We introduce and investigate the injective hull of the strongly Lipschitz classical p-compact operator ideal defined between a pointed metric space and a Banach space. As an application we extend some characterizations of the injective hull of the strongly Lipschitz classical p-compact from the linear case to the Lipschitz case. Also, we introduce the ideal of Lipschitz unconditionally quasi p-nuclear operators between pointed metric spaces and show that it coincides with the Lipschitz injective hull of the ideal of Lipschitz classical p-compact operators.

scholarly journals Bounded approximation properties in terms of $C[0,1]$

2012 ◽  
Vol 110 (1) ◽  
pp. 45 ◽  
Author(s):  
Åsvald Lima ◽  
Vegard Lima ◽  
Eve Oja
Keyword(s):  
Banach Space ◽  
Finite Rank ◽  
Approximation Property ◽  
Integral Operators ◽  
Operator Ideal ◽  
Approximation Properties ◽  
Finite Rank Operators ◽  
Nuclear Operators ◽  
The Ideal

Let $X$ be a Banach space and let $\mathcal I$ be the Banach operator ideal of integral operators. We prove that $X$ has the $\lambda$-bounded approximation property ($\lambda$-BAP) if and only if for every operator $T\in \mathcal I(X,C[0,1]^*)$ there exists a net $(S_\alpha)$ of finite-rank operators on $X$ such that $S_\alpha\to I_X$ pointwise and 26767 \limsup_\alpha\|TS_\alpha\|_{\mathcal I}\leq\lambda\|T\|_{\mathcal I}. 26767 We also prove that replacing $\mathcal I$ by the ideal $\mathcal N$ of nuclear operators yields a condition which is equivalent to the weak $\lambda$-BAP.

SOME RESULTS OF THE -APPROXIMATION PROPERTY FOR BANACH SPACES

2018 ◽  
Vol 61 (03) ◽  
pp. 545-555 ◽  
Author(s):  
JU MYUNG KIM
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Dual Space ◽  
Approximation Property ◽  
Compact Operators ◽  
Operator Ideal ◽  
Formula Group ◽  
Image Position ◽  
The Ideal

AbstractGiven a Banach operator ideal $\mathcal A$, we investigate the approximation property related to the ideal of $\mathcal A$-compact operators, $\mathcal K_{\mathcal A}$-AP. We prove that a Banach space X has the $\mathcal K_{\mathcal A}$-AP if and only if there exists a λ ≥ 1 such that for every Banach space Y and every R ∈ $\mathcal K_{\mathcal A}$(Y, X), $$ \begin{equation} R \in \overline {\{SR : S \in \mathcal F(X, X), \|SR\|_{\mathcal K_{\mathcal A}} \leq \lambda \|R\|_{\mathcal K_{\mathcal A}}\}}^{\tau_{c}}. \end{equation} $$ For a surjective, maximal and right-accessible Banach operator ideal $\mathcal A$, we prove that a Banach space X has the $\mathcal K_{(\mathcal A^{{\rm adj}})^{{\rm dual}}}$-AP if the dual space of X has the $\mathcal K_{\mathcal A}$-AP.

scholarly journals Inequalities for the Schatten p-norm II

1987 ◽  
Vol 29 (1) ◽  
pp. 99-104 ◽  
Author(s):  
Fuad Kittaneh
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Compact Operator ◽  
Trace Class ◽  
Compact Operators ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
The Ideal

This paper is a continuation of [3] in which some inequalities for the Schatten p-norm were considered. The purpose of the present paper is to improve some inequalities in [3] as well as to give more inequalities in the same spirit.Let H be a separable, infinite dimensional complex Hilbert space, and let B(H) denote the algebra of all bounded linear operators acting on H. Let K(H) denote the closed two-sided ideal of compact operators on H. For any compact operator A, let |A| = (A*A)½ and s1(A), s2(A),… be the eigenvalues of |A| in decreasing order and repeated according to multiplicity. A compact operator A is said to be in the Schatten p-class Cp(1 ≤ p < ∞), if Σ s1(A)p < ∞. The Schatten p-norm of A is defined by ∥A∥p = (Σ si(A)p)1/p. This norm makes Cp into a Banach space. Hence C1 is the trace class and C2 is the Hilbert-Schmidt class. It is reasonable to let C∞ denote the ideal of compact operators K(H), and ∥.∥∞ stand for the usual operator norm.

scholarly journals Tight Embeddability of Proper and Stable Metric Spaces

2015 ◽  
Vol 3 (1) ◽  
Author(s):  
F. Baudier ◽  
G. Lancien
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Metric Space ◽  
Metric Spaces ◽  
Proper Subset ◽  
Reflexive Banach Spaces

Abstract We introduce the notions of almost Lipschitz embeddability and nearly isometric embeddability. We prove that for p ∈ [1,∞], every proper subset of Lp is almost Lipschitzly embeddable into a Banach space X if and only if X contains uniformly the ℓpn’s. We also sharpen a result of N. Kalton by showing that every stable metric space is nearly isometrically embeddable in the class of reflexive Banach spaces.

scholarly journals SPACES OF SMALL METRIC COTYPE

2010 ◽  
Vol 02 (04) ◽  
pp. 581-597 ◽  
Author(s):  
E. VEOMETT ◽  
K. WILDRICK
Keyword(s):  
Banach Space ◽  
Metric Space ◽  
Metric Spaces ◽  
Ultrametric Space ◽  
Partial Converse ◽  
Definition Of ◽  
Hausdorff Limits

Mendel and Naor's definition of metric cotype extends the notion of the Rademacher cotype of a Banach space to all metric spaces. Every Banach space has metric cotype at least 2. We show that any metric space that is bi-Lipschitz is equivalent to an ultrametric space having infimal metric cotype 1. We discuss the invariance of metric cotype inequalities under snowflaking mappings and Gromov–Hausdorff limits, and use these facts to establish a partial converse of the main result.

scholarly journals Compact reduction in Lipschitz-free spaces

2021 ◽  
Vol 151 (6) ◽  
pp. 1683-1699
Author(s):  
Ramón J. Aliaga ◽  
Camille Noûs ◽  
Colin Petitjean ◽  
Antonín Procházka
Keyword(s):  
Banach Space ◽  
Metric Space ◽  
General Principle ◽  
Metric Spaces ◽  
Approximation Property ◽  
Complete Metric Space ◽  
Infinite Dimensional ◽  
Schur Property ◽  
Free Spaces ◽  
Compact Subsets

We prove a general principle satisfied by weakly precompact sets of Lipschitz-free spaces. By this principle, certain infinite dimensional phenomena in Lipschitz-free spaces over general metric spaces may be reduced to the same phenomena in free spaces over their compact subsets. As easy consequences we derive several new and some known results. The main new results are: $\mathcal {F}(X)$ is weakly sequentially complete for every superreflexive Banach space $X$, and $\mathcal {F}(M)$ has the Schur property and the approximation property for every scattered complete metric space $M$.

scholarly journals Coarse infinite-dimensionality of hyperspaces of finite subsets

2021 ◽  
Author(s):  
Thomas Weighill ◽  
Takamitsu Yamauchi ◽  
Nicolò Zava
Keyword(s):  
Banach Space ◽  
Metric Space ◽  
Metric Spaces ◽  
Hausdorff Metric ◽  
Convex Banach Space ◽  
Uniformly Convex Banach Space ◽  
Uniformly Convex ◽  
Dimensional Properties ◽  
Bounded Geometry ◽  
Infinite Dimensional

AbstractWe consider infinite-dimensional properties in coarse geometry for hyperspaces consisting of finite subsets of metric spaces with the Hausdorff metric. We see that several infinite-dimensional properties are preserved by taking the hyperspace of subsets with at most n points. On the other hand, we prove that, if a metric space contains a sequence of long intervals coarsely, then its hyperspace of finite subsets is not coarsely embeddable into any uniformly convex Banach space. As a corollary, the hyperspace of finite subsets of the real line is not coarsely embeddable into any uniformly convex Banach space. It is also shown that every (not necessarily bounded geometry) metric space with straight finite decomposition complexity has metric sparsification property.

scholarly journals Best simultaneous approximation on metric spaces via monotonous norms

Filomat ◽  
2020 ◽  
Vol 34 (11) ◽  
pp. 3777-3787
Author(s):  
Mona Khandaqji ◽  
Aliaa Burqan
Keyword(s):  
Banach Space ◽  
Continuous Function ◽  
Finite Number ◽  
Metric Space ◽  
Measure Space ◽  
Simultaneous Approximation ◽  
Metric Spaces ◽  
Closed Subspace ◽  
Integrable Functions ◽  
Best Simultaneous Approximation

For a Banach space X, L?(T,X) denotes the metric space of all X-valued ?-integrable functions f : T ? X, where the measure space (T,?,?) is a complete positive ?-finite and ? is an increasing subadditive continuous function on [0,?) with ?(0) = 0. In this paper we discuss the proximinality problem for the monotonous norm on best simultaneous approximation from the closed subspace Y?X to a finite number of elements in X.

A coarse Mayer–Vietoris principle

1993 ◽  
Vol 114 (1) ◽  
pp. 85-97 ◽  
Author(s):  
Nigel Higson ◽  
John Roe ◽  
Guoliang Yu
Keyword(s):  
Riemannian Manifold ◽  
Metric Space ◽  
Metric Spaces ◽  
Compact Operators ◽  
Cohomology Theory ◽  
Coarse Geometry ◽  
Locally Compact ◽  
Finite Propagation ◽  
Noncompact Riemannian Manifold

In [1], [4], and [6] the authors have studied index problems associated with the ‘coarse geometry’ of a metric space, which typically might be a complete noncompact Riemannian manifold or a group equipped with a word metric. The second author has introduced a cohomology theory, coarse cohomology, which is functorial on the category of metric spaces and coarse maps, and which can be computed in many examples. Associated to such a metric space there is also a C*-algebra generated by locally compact operators with finite propagation. In this note we will show that for suitable decompositions of a metric space there are Mayer–Vietoris sequences both in coarse cohomology and in the K-theory of the C*-algebra. As an application we shall calculate the K-theory of the C*-algebra associated to a metric cone. The result is consistent with the calculation of the coarse cohomology of the cone, and with a ‘coarse’ version of the Baum–Connes conjecture.

scholarly journals Coarse compactifications and controlled products

2020 ◽  
pp. 1-26
Author(s):  
Tomohiro Fukaya ◽  
Shin-ichi Oguni ◽  
Takamitsu Yamauchi
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Equivalence Classes ◽  
Ideal Boundary ◽  
Bijective Correspondence ◽  
Gromov Boundary ◽  
The Ideal

We introduce the notion of controlled products on metric spaces as a generalization of Gromov products, and construct boundaries by using controlled products, which we call the Gromov boundaries. It is shown that the Gromov boundary with respect to a controlled product on a proper metric space is the ideal boundary of a coarse compactification of the space. It is also shown that there is a bijective correspondence between the set of all coarse equivalence classes of controlled products and the set of all equivalence classes of coarse compactifications.

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