The dual space of (L(X,Y),τp) and the p-approximation property

The p-approximation property in terms of density of finite rank operators

Many recent papers have been devoted to the ideal of p-compact operators, denoted by Kp. In this note, we will revisit their main results by very simple proofs that are based on the general theory of operator ideals as developed in the author’s monograph more than 30 years ago. In several cases the outcome is even better. Moreover, the ideal of all operators with p-summing duals is identified as the maximal hull of Kp. As a byproduct, we show that Kmax p is stable under the formation of projective tensor products. Operator ideals play an important role in the study of Banach spaces and their operators. The aim of this note is to demonstrate the well-known interplay of the basic operations on operator ideals. This approach allows us to replace several long reasonings by one-line-proofs. Our methods are applied to the ideal Kp formed by the p-compact operators. The most significant observation says that this ideal is surjective and regular. Moreover, we show that the maximal hull of Kp consists of all operators whose dual is p-summing. For unexplained notation and terminology in the text we refer to [15]; auxiliary results, adopted from [15], are just marked by triplets, such as (4.7.16), which means [15, 4.7.16]. For simplicity, we mostly deal with operator ideals A, B, . . . only, but not with the underlying norms A, B, . . . . It should be noted, however, that (in our context) A=B always includes A=B. According to [19, p. 20], a subset K of a Banach space E is called relatively p-compact if there exists a sequence (xn) in E such that K ⊆ { ∞ ∑

https://www.ams.org/proc/2014-142-02/S0002-9939-2013-11758-2/S0002-9939-2013-11758-2.pdf

The ideal of p -compact operators and its maximal hull

Lipschitz operator ideals and the approximation property

Density of finite rank operators in the Banach space of p-compact operators

Let X and Y be Banach spaces. We say that a set M ⊂ X(X, Y) (K(X, Y) denotes the space of all compact operators from X into Y) is equicompact if there exists a null sequence (x* n ) n in X* such that ||Tx|| ≤ sup n |x* n (x)| for all x ∈ X and all T ∈ M. It is easy to show that collectively compactness and equicompactness are dual concepts in the following sense: M is equicompact iff M* = {T*: T ∈ M} is collectively compact. We study some properties of equicompact sets and, among other results, we prove: 1) a set M C K(X,Y) is equicompact iff each bounded sequence (x n ) n in X has a subsequence (x k(n) ) n such that (Tx k(n) ) n is a converging sequence uniformly for T ∈ M; 2) if Y does not have finite cotype and M C X(X, Y) is a maximal equicompact set, then, given e > 0 and a finite set {x 1 ,...,x n } in X, there is an operator S ∈ M such that ||Tx i || < (1 + e)||Sx i || for i = 1,...,n and all T ∈ M.

/pdf/equicompact-sets-of-operators-defined-on-banach-spaces-30ufysd4ed.pdf

Equicompact sets of operators defined on Banach spaces

Let X and Y be Banach spaces. We say that a set \(M \subset \mathcal{W}(X,Y)\) (the space of all weakly compact operators from X into Y) is weakly equicompact if, for every bounded sequence (xn) in X, there exists a subsequence (xk(n)) so that (Txk(n)) is weakly uniformly convergent for T ∈ M. We study some properties of weakly equicompact sets and, among other results, we prove: 1) if \(M \subset \mathcal{W}(X,Y)\) is collectively weakly compact, then M* is weakly equicompact iff M**x**={T**x** : T ∈ M} is relatively compact in Y for every x** ∈X**; 2) weakly equicompact sets are precompact in \(\mathcal{L}(X,Y)\) for the topology of uniform convergence on the weakly null sequences in X.

Weakly equicompact sets of operators defined on Banach spaces

Let X and Y be Banach spaces. A subset M of K(X, Y ) (the vector space of all compact operators from X into Y endowed with the operator norm) is said to be equicompact if every bounded sequence (xn) in X has a subsequence (xk(n))n such that (Txk(n))n is uniformly convergent for T ∈ M. We study the relationship between this concept and the notion of uniformly completely continuous set and give some applications. Among other results, we obtain a generalization of the classical Ascoli theorem and a compactness criterion in Mc(F, X), the Banach space of all (finitely additive) vector measures (with compact range) from a field F of sets into X endowed with the semivariation norm.

/pdf/some-properties-and-applications-of-equicompact-sets-of-qmssbh6c80.pdf

Some properties and applications of equicompact sets of operators

Handle everyday research tasks with reliable, citation-backed results

Your personal Research Agent to handle research tasks with citation-backed results

Popular Tasks used by Researchers

How can I help with your research?

Meet SciSpace

Get more enhanced response by uploading the PDFs you want me to reference.

No relevant PDFs in your library

SciSpace is the AI research assistant for academics. Run systematic literature reviews on 280M+ papers, and write papers with cited sources. Trusted by 1M+ students, PhDs & researchers.

SciSpace | AI for Research

Analyze PDFs

Code & Manuscripts

Funding & Grants

Literature & Patents

Medical & Clinical Data

Systematic Review

Visualize & Present

Web & Data

Build a Google Scholar-like website for your research.

Build a website

Create charts and images for your research

Create a Chart

Write a paper for submission to a journal

Draft a manuscript

Patent Search

Design eye-catching scientific posters in minutes.

Scientific Poster Generation

Systematic Literature Review

One task is running at the moment. Your messages will be shown right after.

Drag and drop or click here to browse

Loved by <highlight>1 million+</highlight> researchers

Extract a list of specific topics and their sources from unstructured text

Topics

Compare and analyze relevant papers that matches with your search

Papers

Get insights from PDFs and bookmarked papers from your library

My library

Recent searches

Try searching for:

Catch AI-generated content in scholarly and non-scholarly content

{ai} Detector

Ai Writer

Get PDF Summaries, highlighted text explanations 

Chat with PDF

Effortlessly create in-text citations and bibliographies in APA and 2,500 other formats

Citation generator

Get explanations, summaries, and answers on academic papers

Ease up your research workflow with {scispace}'s cohort of exciting AI tools

Elevate your academic writing skills and convey your ideas the way you want

Paraphraser

Explore our range of reading and writing tools

Your file is being prepared and should be ready in a few minutes. If it's a large file, it might take a bit longer. You can close this window, and we'll email you the file when it's done.

You have reached a maximum limit of <strong>{limit}</strong> columns in the table. Remove at least <strong>1</strong> column to add or create another one.

E. Serrano

Author Tools

Chat about Author