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.

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 Pre-Quasi Simple Banach Operator Ideal Generated by s−Numbers

2020 ◽  
Vol 2020 ◽  
pp. 1-11 ◽  
Author(s):  
Awad A. Bakery ◽  
Afaf R. Abou Elmatty
Keyword(s):  
Banach Space ◽  
Sequence Space ◽  
Finite Rank ◽  
Necessary Conditions ◽  
Operator Ideal ◽  
Approximation Numbers ◽  
Weighted Mean ◽  
Finite Rank Operators ◽  
Generalized Cesáro Sequence Space

Let E be a weighted Nakano sequence space or generalized Cesáro sequence space defined by weighted mean and by using s−numbers of operators from a Banach space X into a Banach space Y. We give the sufficient (not necessary) conditions on E such that the components SEX,Y≔T∈LX,Y:snTn=0∞∈E of the class SE form pre-quasi operator ideal, the class of all finite rank operators are dense in the Banach pre-quasi ideal SE, the pre-quasi operator ideal formed by the sequence of approximation numbers is strictly contained for different weights and powers, the pre-quasi Banach Operator ideal formed by the sequence of approximation numbers is small, and finally, the pre-quasi Banach operator ideal constructed by s−numbers is simple Banach space.

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 Homological characterizations of the approximation property for Banach spaces

1992 ◽  
Vol 34 (2) ◽  
pp. 229-239 ◽  
Author(s):  
Yu. V. Selivanov
Keyword(s):  
Banach Space ◽  
Banach Algebra ◽  
Banach Spaces ◽  
Cohomology Group ◽  
Approximation Property ◽  
Three Dimensional ◽  
Nuclear Operators ◽  
Finite Dimensionality

Let E be a Banach space, and let N(E) be the Banach algebra of all nuclear operators on E. In this work, we shall study the homological properties of this algebra. Some of these properties turn out to be equivalent to the (Grothendieck) approximation property for E. These include:(i) biprojectivity of N(E);(ii) biflatness of N(E);(iii) homological finite-dimensionality of N(E);(iv) vanishing of the three-dimensional cohomology group, H3(N(E), N(E)).

Range Inclusion for Multilinear Mappings: Applications

1985 ◽  
Vol 28 (3) ◽  
pp. 317-320
Author(s):  
C. K. Fong
Keyword(s):  
Hilbert Space ◽  
Finite Rank ◽  
Trace Class ◽  
Inner Derivation ◽  
Finite Rank Operators ◽  
Multilinear Mappings ◽  
Trace Class Operators ◽  
The Ideal

AbstractThe result of S. Grabiner [5] on range inclusion is applied for establishing the following two theorems: 1. For A, B ∊ L(H), two operators on the Hilbert space H, we have DBC0(H) ⊆ DAL(H) if and only if DBC1(H) ⊆ DAL(H), where DA is the inner derivation which sends S ∊ L(H) to AS - SA, C1(H) is the ideal of trace class operators and C0(H) is the ideal of finite rank operators. 2. (Due to Fialkow [3]) For A, B ∊ L(H), we write T(A, B) for the map on L(H) sending S to AS - SB. Then the range of T(A, B)is the whole L(H) if it includes all finite rank operators L(H).

scholarly journals Strongly Extreme Points and Approximation Properties

2018 ◽  
Vol 61 (3) ◽  
pp. 449-457
Author(s):  
Trond A. Abrahamsen ◽  
Petr Hájek ◽  
Olav Nygaard ◽  
Stanimir L. Troyanski
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Extreme Point ◽  
Convex Subset ◽  
Approximation Property ◽  
Extreme Points ◽  
Local Approximation ◽  
Approximation Properties ◽  
Compact Approximation ◽  
Approximation Of The Identity

AbstractWe show that if x is a strongly extreme point of a bounded closed convex subset of a Banach space and the identity has a geometrically and topologically good enough local approximation at x, then x is already a denting point. It turns out that such an approximation of the identity exists at any strongly extreme point of the unit ball of a Banach space with the unconditional compact approximation property. We also prove that every Banach space with a Schauder basis can be equivalently renormed to satisfy the suõcient conditions mentioned.

scholarly journals Computability-theoretic complexity of effective Banach spaces

2022 ◽  
Author(s):  
◽  
Long Qian
Keyword(s):  
Banach Space ◽  
Lower Bound ◽  
Banach Spaces ◽  
Lower Bounds ◽  
Approximation Property ◽  
Upper Bounds ◽  
Space Theory ◽  
Approximation Properties ◽  
Open Problems

<p><b>We investigate the geometry of effective Banach spaces, namely a sequenceof approximation properties that lies in between a Banach space having a basis and the approximation property.</b></p> <p>We establish some upper bounds on suchproperties, as well as proving some arithmetical lower bounds. Unfortunately,the upper bounds obtained in some cases are far away from the lower bound.</p> <p>However, we will show that much tighter bounds will require genuinely newconstructions, and resolve long-standing open problems in Banach space theory.</p> <p>We also investigate the effectivisations of certain classical theorems in Banachspaces.</p>

scholarly journals Some properties of pre-quasi operator ideal of type generalized Cesáro sequence space defined by weighted means

2019 ◽  
Vol 17 (1) ◽  
pp. 1703-1715 ◽  
Author(s):  
Awad A. Bakery ◽  
Mustafa M. Mohammed
Keyword(s):  
Banach Space ◽  
Sequence Space ◽  
Finite Rank ◽  
Necessary Conditions ◽  
Operator Ideal ◽  
Linear Operators ◽  
Approximation Numbers ◽  
Bounded Linear ◽  
Weighted Means ◽  
Generalized Cesáro Sequence Space

Abstract Let E be a generalized Cesáro sequence space defined by weighted means and by using s-numbers of operators from a Banach space X into a Banach space Y. We give the sufficient (not necessary) conditions on E such that the components $$\begin{array}{} \displaystyle S_{E}(X, Y):=\Big\{T\in L(X, Y):((s_{n}(T))_{n=0}^{\infty}\in E\Big\}, \end{array}$$ of the class SE form pre-quasi operator ideal, the class of all finite rank operators are dense in the Banach pre-quasi ideal SE, the pre-quasi operator ideal formed by the sequence of approximation numbers is strictly contained for different weights and powers, the pre-quasi Banach Operator ideal formed by the sequence of approximation numbers is small and the pre-quasi Banach operator ideal constructed by s-numbers is simple Banach space. Finally the pre-quasi operator ideal formed by the sequence of s-numbers and this sequence space is strictly contained in the class of all bounded linear operators, whose sequence of eigenvalues belongs to this sequence space.

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.

scholarly journals A Trace Formula for the Index of B-Fredholm Operators

2018 ◽  
Vol 61 (4) ◽  
pp. 1063-1068 ◽  
Author(s):  
Mohammed Berkani
Keyword(s):  
Banach Space ◽  
Banach Algebra ◽  
Fredholm Operator ◽  
Trace Formula ◽  
Finite Rank ◽  
Toeplitz Operators ◽  
Drazin Inverse ◽  
Trace Function ◽  
Fredholm Operators ◽  
The Ideal

AbstractIn this paper we define B-Fredholm elements in a Banach algebraAmodulo an idealJofA. When a trace function is given on the idealJ, it generates an index for B-Fredholm elements. In the case of a B-Fredholm operatorTacting on a Banach space, we prove that its usual index ind(T) is equal to the trace of the commutator [T, T0], whereT0is a Drazin inverse ofTmodulo the ideal of finite rank operators, extending Fedosov's trace formula for Fredholm operators (see Böttcher and Silbermann [Analysis of Toeplitz operators, 2nd edn (Springer, 2006)]. In the case of a primitive Banach algebra, we prove a punctured neighbourhood theorem for the index.

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