scholarly journals Norm Attaining Arens Extensions onℓ1

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Javier Falcó ◽  
Domingo García ◽  
Manuel Maestre ◽  
Pilar Rueda
Keyword(s):  
Banach Spaces ◽  
A Priori ◽  
Multilinear Form ◽  
Multilinear Forms ◽  
Norm Attaining

We study norm attaining properties of the Arens extensions of multilinear forms defined on Banach spaces. Among other related results, we construct a multilinear form onℓ1with the property that only some fixed Arens extensions determined a priori attain their norms. We also study when multilinear forms can be approximated by ones with the property that only some of their Arens extensions attain their norms.

scholarly journals $B$-CONVEX OPERATOR SPACES

2003 ◽  
Vol 46 (3) ◽  
pp. 649-668 ◽  
Author(s):  
Javier Parcet
Keyword(s):  
Banach Spaces ◽  
A Priori ◽  
Operator Space ◽  
Subject Classification ◽  
Operator Spaces ◽  
Secondary 42C15 ◽  
Convex Operator ◽  
Primary 46L07

AbstractThe notion of $B$-convexity for operator spaces, which a priori depends on a set of parameters indexed by $\sSi$, is defined. Some of the classical characterizations of this geometric notion for Banach spaces are studied in this new context. For instance, an operator space is $B_{\sSi}$-convex if and only if it has $\sSi$-subtype. The class of uniformly non-$\mathcal{L}^1(\sSi)$ operator spaces, which is also the class of $B_{\sSi}$-convex operator spaces, is introduced. Moreover, an operator space having non-trivial $\sSi$-type is $B_{\sSi}$-convex. However, the converse is false. The row and column operator spaces are nice counterexamples of this fact, since both are Hilbertian. In particular, this result shows that a version of the Maurey–Pisier Theorem does not hold in our context. Some other examples of Hilbertian operator spaces will be considered. In the last part of this paper, the independence of $B_{\sSi}$-convexity with respect to $\sSi$ is studied. This provides some interesting problems, which will be posed.AMS 2000 Mathematics subject classification: Primary 46L07. Secondary 42C15

scholarly journals A NOTE ON L2-SUMMAND VECTORS IN DUAL SPACES

2008 ◽  
Vol 50 (3) ◽  
pp. 429-432 ◽  
Author(s):  
ANTONIO AIZPURU ◽  
FRANCISCO J GARCÍA-PACHECO
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Real Banach Space ◽  
Dual Spaces ◽  
Equivalent Norms ◽  
Norm Attaining

AbstractIt is shown that every L2-summand vector of a dual real Banach space is a norm-attaining functional. As consequences, the L2-summand vectors of a dual real Banach space can be determined by the L2-summand vectors of its predual; for every n ∈ , every real Banach space can be equivalently renormed so that the set of norm-attaining functionals is n-lineable; and it is easy to find equivalent norms on non-reflexive dual real Banach spaces that are not dual norms.

scholarly journals Coincidence Results for Summing Multilinear Mappings

2016 ◽  
Vol 59 (4) ◽  
pp. 877-897 ◽  
Author(s):  
Oscar Blasco ◽  
Geraldo Botelho ◽  
Daniel Pellegrino ◽  
Pilar Rueda
Keyword(s):  
Banach Spaces ◽  
A Priori ◽  
New Techniques ◽  
Multilinear Mappings

AbstractIn this paper we prove coincidence results concerning spaces of absolutely summing multilinear mappings between Banach spaces. The nature of these results arises from two distinct approaches: the coincidence of two a priori different classes of summing multilinear mappings, and the summability of all multilinear mappings defined on products of Banach spaces. Optimal generalizations of known results are obtained. We also introduce and explore new techniques in the field: for example, a technique to extend coincidence results for linear, bilinear and even trilinear mappings to general multilinear ones.

scholarly journals Norm Attaining Multilinear Forms on

2008 ◽  
Vol 2008 ◽  
pp. 1-6
Author(s):  
Yousef Saleh
Keyword(s):  
Banach Space ◽  
Linear Operators ◽  
Bilinear Forms ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Multilinear Forms ◽  
Arbitrary Measure ◽  
Norm Attaining Operators ◽  
Norm Attaining

Given an arbitrary measure , this study shows that the set of norm attaining multilinear forms is not dense in the space of all continuous multilinear forms on . However, we have the density if and only if is purely atomic. Furthermore, the study presents an example of a Banach space in which the set of norm attaining operators from into is dense in the space of all bounded linear operators . In contrast, the set of norm attaining bilinear forms on is not dense in the space of continuous bilinear forms on .

scholarly journals The Ivanov regularized Gauss–Newton method in Banach space with an a posteriori choice of the regularization radius

2019 ◽  
Vol 27 (4) ◽  
pp. 539-557
Author(s):  
Barbara Kaltenbacher ◽  
Andrej Klassen ◽  
Mario Luiz Previatti de Souza
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Newton Method ◽  
Noise Level ◽  
Numerical Experiments ◽  
Convergence Rates ◽  
A Priori ◽  
Discrepancy Principle ◽  
A Posteriori ◽  
Source Condition

Abstract In this paper, we consider the iteratively regularized Gauss–Newton method, where regularization is achieved by Ivanov regularization, i.e., by imposing a priori constraints on the solution. We propose an a posteriori choice of the regularization radius, based on an inexact Newton/discrepancy principle approach, prove convergence and convergence rates under a variational source condition as the noise level tends to zero and provide an analysis of the discretization error. Our results are valid in general, possibly nonreflexive Banach spaces, including, e.g., {L^{\infty}} as a preimage space. The theoretical findings are illustrated by numerical experiments.

Error estimates for the simplified iteratively regularized Gauss–Newton method in Banach spaces under a Morozov-type stopping rule

2018 ◽  
Vol 26 (3) ◽  
pp. 311-333 ◽  
Author(s):  
Pallavi Mahale ◽  
Sharad Kumar Dixit
Keyword(s):  
Banach Spaces ◽  
Error Estimates ◽  
Newton Method ◽  
A Priori ◽  
Identification Problem ◽  
Stopping Rule ◽  
Optimal Error Estimates ◽  
Optimal Error ◽  
Parameter Identification Problem ◽  
Ill Posed

AbstractJin Qinian and Min Zhong [10] considered an iteratively regularized Gauss–Newton method in Banach spaces to find a stable approximate solution of the nonlinear ill-posed operator equation. They have considered a Morozov-type stopping rule (Rule 1) as one of the criterion to stop the iterations and studied the convergence analysis of the method. However, no error estimates have been obtained for this case. In this paper, we consider a modified variant of the method, namely, the simplified Gauss–Newton method under both an a priori as well as a Morozov-type stopping rule. In both cases, we obtain order optimal error estimates under Hölder-type approximate source conditions. An example of a parameter identification problem for which the method can be implemented is discussed in the paper.

scholarly journals A converse result for Banach space convergence rates in Tikhonov-type convex regularization of ill-posed linear equations

2018 ◽  
Vol 26 (5) ◽  
pp. 639-646 ◽  
Author(s):  
Jens Flemming
Keyword(s):  
Banach Spaces ◽  
Convergence Rates ◽  
A Priori ◽  
Linear Equations ◽  
Rate Function ◽  
Bregman Distances ◽  
Error Measures ◽  
Ill Posed ◽  
General Convex ◽  
Linear Operator Equations

Abstract We consider Tikhonov-type variational regularization of ill-posed linear operator equations in Banach spaces with general convex penalty functionals. Upper bounds for certain error measures expressing the distance between exact and regularized solutions, especially for Bregman distances, can be obtained from variational source conditions. We prove that such bounds are optimal in case of twisted Bregman distances, a specific a priori parameter choice, and low regularity of the exact solution, that is, the rate function is also an asymptotic lower bound for the error measure. This result extends existing converse results from Hilbert space settings to Banach spaces without adhering to spectral theory.

scholarly journals Global existence for abstract nonlinear Volterra–Fredholm functional integrodifferential equation

2012 ◽  
Vol 45 (1) ◽  
Author(s):  
M. B. Dhakne ◽  
Kishor D. Kucche
Keyword(s):  
Global Existence ◽  
Banach Spaces ◽  
Integrodifferential Equation ◽  
Initial Value Problem ◽  
Existence Of Solutions ◽  
A Priori ◽  
A Priori Bounds ◽  
Initial Value ◽  
Global Existence Of Solutions ◽  
Fredholm Functional

AbstractIn the present paper, we investigate the global existence of solutions to initial value problem for nonlinear mixed Volterra–Fredholm functional integrodifferential equations in Banach spaces. The technique used in our analysis is based on an application of the topological transversality theorem known as Leray–Schauder alternative and rely on a priori bounds of solution.

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