scholarly journals The free Banach lattice generated by a Banach space

2018 ◽  
Vol 274 (10) ◽  
pp. 2955-2977 ◽  
Author(s):  
Antonio Avilés ◽  
José Rodríguez ◽  
Pedro Tradacete
Keyword(s):  
Banach Space ◽  
Banach Lattice

Tauberian theorems in a Banach lattice, with applications to the LP spaces

1954 ◽  
Vol 50 (2) ◽  
pp. 242-249
Author(s):  
D. C. J. Burgess
Keyword(s):  
Banach Space ◽  
Banach Lattice ◽  
Laplace Transforms ◽  
Tauberian Theorems ◽  
General Argument ◽  
Arbitrary Banach Space ◽  
Lp Spaces ◽  
Special Case

In a previous paper (2) of the author, there was given a treatment of Tauberian theorems for Laplace transforms with values in an arbitrary Banach space. Now, in § 2 of the present paper, this kind of technique is applied to the more special case of Laplace transforms with values in a Banach lattice, and investigations are made on what additional results can be obtained by taking into account the existence of an ordering relation in the space. The general argument is again based on Widder (5) to which frequent references are made.

scholarly journals Banach spaces with property (w)

1993 ◽  
Vol 35 (2) ◽  
pp. 207-217 ◽  
Author(s):  
Denny H. Leung
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Banach Lattice ◽  
Banach Lattices ◽  
Weakly Compact ◽  
Type Spaces

A Banach space E is said to have Property (w) if every operator from E into E' is weakly compact. This property was introduced by E. and P. Saab in [9]. They observe that for Banach lattices, Property (w) is equivalent to Property (V*), which in turn is equivalent to the Banach lattice having a weakly sequentially complete dual. Thus the following question was raised in [9].Does every Banach space with Property (w) have a weakly sequentially complete dual, or even Property (V*)?In this paper, we give two examples, both of which answer the question in the negative. Both examples are James type spaces considered in [1]. They both possess properties stronger than Property (w). The first example has the property that every operator from the space into the dual is compact. In the second example, both the space and its dual have Property (w). In the last section we establish some partial results concerning the problem (also raised in [9]) of whether (w) passes from a Banach space E to C(K, E).

scholarly journals Positive p-summing operators, vector measures and tensor products

1988 ◽  
Vol 31 (2) ◽  
pp. 179-184 ◽  
Author(s):  
Oscar Blasco
Keyword(s):  
Banach Space ◽  
Tensor Product ◽  
Banach Lattice ◽  
Tensor Products ◽  
Boundary Values ◽  
Vector Measures ◽  
Absolutely Summing Operators

In this paper we shall introduce a certain class of operators from a Banach lattice X into a Banach space B (see Definition 1) which is closely related to p-absolutely summing operators defined by Pietsch [8].These operators, called positive p-summing, have already been considered in [9] in the case p = 1 (there they are called cone absolutely summing, c.a.s.) and in [1] by the author who found this space to be the space of boundary values of harmonic B-valued functions in .Here we shall use these spaces and the space of majorizing operators to characterize the space of bounded p-variation measures and to endow the tensor product with a norm in order to get as its completion in this norm.

scholarly journals Geometric Characterization of Injective Banach Lattices

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 250
Author(s):  
Anatoly Kusraev ◽  
Semën Kutateladze
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Banach Spaces ◽  
Set Theory ◽  
Banach Lattice ◽  
Dual Space ◽  
Banach Lattices ◽  
Transfer Principle ◽  
Ordered Banach Spaces

This is a continuation of the authors’ previous study of the geometric characterizations of the preduals of injective Banach lattices. We seek the properties of the unit ball of a Banach space which make the space isometric or isomorphic to an injective Banach lattice. The study bases on the Boolean valued transfer principle for injective Banach lattices. The latter states that each such lattice serves as an interpretation of an AL-space in an appropriate Boolean valued model of set theory. External identification of the internal Boolean valued properties of the corresponding AL-spaces yields a characterization of injective Banach lattices among Banach spaces and ordered Banach spaces. We also describe the structure of the dual space and present some dual characterization of injective Banach lattices.

scholarly journals Upper separated multifunctions in deterministic and stochastic optimal control

2017 ◽  
Vol 2 (2) ◽  
pp. 479-484 ◽  
Author(s):  
Jerzy Motyl
Keyword(s):  
Banach Space ◽  
Optimal Control ◽  
Banach Lattice ◽  
Stochastic Optimal Control ◽  
Optimal Control Problems ◽  
Existence Of Solutions ◽  
Control Problems

AbstractLet X be a Banach space while (Y,⪯) a Banach lattice. We consider the class of “upper separated” set-valued functions F : X → 2Y and investigate the problem of the existence of order-convex selections of F. First, we present results on the existence of the Carathéodory-convex type selections of upper separated multifunctions and apply them to investigation of the existence of solutions of differential and stochastic inclusions. We will discuss the applicability of obtained selection results to some deterministic and stochastic optimal control problems.

scholarly journals A quantitative approach to disjointly non-singular operators

2021 ◽  
Vol 115 (4) ◽  
Author(s):  
Manuel González ◽  
Antonio Martinón
Keyword(s):  
Banach Space ◽  
Banach Lattice ◽  
Quantitative Approach ◽  
Singular Operators ◽  
Strictly Singular ◽  
Operational Quantities ◽  
Order Continuous

AbstractWe introduce and study some operational quantities which characterize the disjointly non-singular operators from a Banach lattice E to a Banach space Y when E is order continuous, and some other quantities which characterize the disjointly strictly singular operators for arbitrary E.

Korovkin Theorems for Integral Operators with Kernels of Finite Oscillation

1974 ◽  
Vol 26 (6) ◽  
pp. 1390-1404 ◽  
Author(s):  
M. J. Marsden ◽  
S. D. Riemenschneider
Keyword(s):  
Banach Space ◽  
Banach Lattice ◽  
Measure Space ◽  
Integral Operators ◽  
Finite Measure ◽  
Bounded Sequence ◽  
Positive Operators ◽  
Linear Operators ◽  
Finite Measure Space

There has been considerable interest recently in the investigation of "Korovkin sets". Briefly, for X a Banach space and a family of linear operators on X, a subset K ⊂ X is a Korovkin set relative to if for any bounded sequence {Tn} ⊂ , Tnk → k in X for each k ∊ K implies Tnx → x for each x ∊ X. A large portion of these investigations have been carried out for X being one of the spaces C(S), S compact Hausdorff, the usual Lp spaces of functions on some finite measure space, or some Banach lattice; while is one of the classes +-positive operators, 1-contractions (i.e., ||T|| 1), or + ⋂1

scholarly journals Representation of p-Lattice Summing Operators

1992 ◽  
Vol 35 (2) ◽  
pp. 267-277 ◽  
Author(s):  
Beatriz Porras Pomares
Keyword(s):  
Banach Space ◽  
Banach Lattice ◽  
Representing Measure

AbstractIn this paper we study some aspects of the behaviour of p-lattice summing operators. We prove first that an operator T from a Banach space E to a Banach lattice X is p-lattice summing if and only if its bitranspose is. Using this theorem we prove a characterization for 1 -lattice summing operators defined on a C(K) space by means of the representing measure, which shows that in this case 1 -lattice and ∞-lattice summing operators coincide. We present also some results for the case 1 ≤ p < ∞ on C(K,E).

Perron-Frobenius and Krein-Rutman theorems for tangentially positive operators

2012 ◽  
Vol 10 (6) ◽  
Author(s):  
Adam Kanigowski ◽  
Wojciech Kryszewski
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Banach Lattice ◽  
Spectral Properties ◽  
Lattice Structure ◽  
Positive Operators ◽  
Unbounded Linear Operator ◽  
Underlying Space ◽  
Spectral Bounds

AbstractWe study several aspects of a generalized Perron-Frobenius and Krein-Rutman theorems concerning spectral properties of a (possibly unbounded) linear operator on a cone in a Banach space. The operator is subject to the so-called tangency or weak range assumptions implying the resolvent invariance of the cone. The further assumptions rely on relations between the spectral and essential spectral bounds of the operator. In general we do not assume that the cone induces the Banach lattice structure into the underlying space.

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