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.

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 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

scholarly journals A class of Banach spaces with few non-strictly singular operators

2005 ◽  
Vol 222 (2) ◽  
pp. 306-384 ◽  
Author(s):  
S.A. Argyros ◽  
J. Lopez-Abad ◽  
S. Todorcevic
Keyword(s):  
Banach Spaces ◽  
Singular Operators ◽  
Strictly Singular

scholarly journals Lebesgue's Differentiation Theorems in R.I. Quasi-Banach Spaces and Lorentz SpacesΓp,w

2012 ◽  
Vol 2012 ◽  
pp. 1-28 ◽  
Author(s):  
Maciej Ciesielski ◽  
Anna Kamińska
Keyword(s):  
Banach Space ◽  
Weight Function ◽  
Banach Spaces ◽  
Maximal Function ◽  
Measurable Function ◽  
Pointwise Convergence ◽  
Lorentz Spaces ◽  
Best Constant ◽  
Decreasing Rearrangement ◽  
Order Continuous

The paper is devoted to investigation of new Lebesgue's type differentiation theorems (LDT) in rearrangement invariant (r.i.) quasi-Banach spacesEand in particular on Lorentz spacesΓp,w={f:∫(f**)pw<∞}for any0<p<∞and a nonnegative locally integrable weight functionw, wheref**is a maximal function of the decreasing rearrangementf*for any measurable functionfon(0,α), with0<α≤∞. The first type of LDT in the spirit of Stein (1970), characterizes the convergence of quasinorm averages off∈E, whereEis an order continuous r.i. quasi-Banach space. The second type of LDT establishes conditions for pointwise convergence of the best or extended best constant approximantsfϵoff∈Γp,worf∈Γp-1,w,1<p<∞, respectively. In the last section it is shown that the extended best constant approximant operator assumes a unique constant value for any functionf∈Γp-1,w,1<p<∞.

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