scholarly journals The history of a general criterium on spaceability

2017 ◽  
Vol 15 (1) ◽  
pp. 252-260
Author(s):  
Víctor M. Sánchez
Keyword(s):  
Banach Spaces ◽  
Operator Ideals ◽  
Survey Paper ◽  
Arbitrary Operator ◽  
History Of

Abstract There are just a few general criteria on spaceability. This survey paper is the history of one of the first ones. Let I1 and I2 be arbitrary operator ideals and E and F be Banach spaces. The spaceability of the set of operators I1(E, F)\ I2(E, F) is studied. Before stating the criterium, the paper summarizes the main results about lineability and spaceability of differences between particular operator ideals obtained in recent years. They are the seed of the ideas contained in the general criterium.

Injective Dual Banach Spaces and Operator Ideals

2021 ◽  
Vol 18 (1) ◽  
Author(s):  
Raffaella Cilia ◽  
Joaquín M. Gutiérrez
Keyword(s):  
Banach Spaces ◽  
Operator Ideals

Duality of the distance to closed operator ideals

2003 ◽  
Vol 133 (1) ◽  
pp. 197-212 ◽  
Author(s):  
Hans-Olav Tylli
Keyword(s):  
Banach Spaces ◽  
Closed Operator ◽  
Operator Ideal ◽  
Linear Operators ◽  
Distance Functions ◽  
Approximation Properties ◽  
Operator Ideals ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Special Operator

Special operator-ideal approximation properties (APs) of Banach spaces are employed to solve the problem of whether the distance functions S ↦ dist(S*, I(F*, E*)) and S ↦ dist(S, I*(E, F)) are uniformly comparable in each space L(E, F) of bounded linear operators. Here, I*(E, F) = {S ∈ L(E, F) : S* ∈ I(F*, E*)} stands for the adjoint ideal of the closed operator ideal I for Banach spaces E and F. Counterexamples are obtained for many classical surjective or injective Banach operator ideals I by solving two resulting ‘asymmetry’ problems for these operator-ideal APs.

scholarly journals Ornstein–Uhlenbeck semigroups in infinite dimension

2020 ◽  
Vol 378 (2185) ◽  
pp. 20190620
Author(s):  
A. Lunardi ◽  
D. Pallara
Keyword(s):  
Banach Spaces ◽  
Hilbert Spaces ◽  
Infinite Dimension ◽  
Theme Issue ◽  
Survey Paper ◽  
Wiener Spaces

This is a survey paper about Ornstein–Uhlenbeck semigroups in infinite dimension and their generators. We start from the classical Ornstein–Uhlenbeck semigroup on Wiener spaces and then discuss the general case in Hilbert spaces. Finally, we present some results for Ornstein–Uhlenbeck semigroups on Banach spaces. This article is part of the theme issue ‘Semigroup applications everywhere’.

scholarly journals On symmetric ideals of multilinear mappings between Banach spaces

2006 ◽  
Vol 81 (1) ◽  
pp. 141-148 ◽  
Author(s):  
Geraldo Botelho ◽  
Daniel M. Pellegrino
Keyword(s):  
Banach Spaces ◽  
Operator Ideals ◽  
Multilinear Mappings

AbstractIn this paper we provide examples and counterexamples of symmetric ideals of multilinear mappings between Banach spaces and prove that if I1, …, In are operator ideals, then the ideals of multilinear mappings L(I1, …, In) and /I1, …, In/ are symmetric if and only if I1 = … = In.

Modern statistical estimation via oracle inequalities

Acta Numerica ◽  
2006 ◽  
Vol 15 ◽  
pp. 257-325 ◽  
Author(s):  
Emmanuel J. Candès
Keyword(s):  
Statistical Estimation ◽  
Applied Mathematics ◽  
Theoretical Computer Science ◽  
Efficient Estimation ◽  
Oracle Inequalities ◽  
Theoretical Computer ◽  
Survey Paper ◽  
Applied Harmonic Analysis ◽  
Potential Applications ◽  
History Of

A number of fundamental results in modern statistical theory involve thresholding estimators. This survey paper aims at reconstructing the history of how thresholding rules came to be popular in statistics and describing, in a not overly technical way, the domain of their application. Two notions play a fundamental role in our narrative: sparsity and oracle inequalities. Sparsity is a property of the object to estimate, which seems to be characteristic of many modern problems, in statistics as well as applied mathematics and theoretical computer science, to name a few. ‘Oracle inequalities’ are a powerful decision-theoretic tool which has served to understand the optimality of thresholding rules, but which has many other potential applications, some of which we will discuss.Our story is also the story of the dialogue between statistics and applied harmonic analysis. Starting with the work of Wiener, we will see that certain representations emerge as being optimal for estimation. A leitmotif throughout our exposition is that efficient representations lead to efficient estimation.

scholarly journals Approximation properties of tensor norms and operator ideals for Banach spaces

2020 ◽  
Vol 18 (1) ◽  
pp. 1698-1708
Author(s):  
Ju Myung Kim
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Approximation Property ◽  
Approximation Properties ◽  
Operator Ideals ◽  
Finitely Generated ◽  
Tensor Norm ◽  
Tensor Norms

Abstract For a finitely generated tensor norm α \alpha , we investigate the α \alpha -approximation property ( α \alpha -AP) and the bounded α \alpha -approximation property (bounded α \alpha -AP) in terms of some approximation properties of operator ideals. We prove that a Banach space X has the λ \lambda -bounded α p , q {\alpha }_{p,q} -AP ( 1 ≤ p , q ≤ ∞ , 1 / p + 1 / q ≥ 1 ) (1\le p,q\le \infty ,1/p+1/q\ge 1) if it has the λ \lambda -bounded g p {g}_{p} -AP. As a consequence, it follows that if a Banach space X has the λ \lambda -bounded g p {g}_{p} -AP, then X has the λ \lambda -bounded w p {w}_{p} -AP.

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