A Geometrical Characterization of Banach Spaces in Which Martingale Difference Sequences are Unconditional

2011 ◽  
pp. 360-374
Author(s):  
Burgess Davis ◽  
Renming Song
Keyword(s):  
Banach Spaces ◽  
Martingale Difference ◽  
Geometrical Characterization ◽  
Difference Sequences

scholarly journals Unconditional martingale difference sequences in Banach spaces

2007 ◽  
Vol 326 (2) ◽  
pp. 1291-1309 ◽  
Author(s):  
Stuart F. Cullender ◽  
Coenraad C.A. Labuschagne
Keyword(s):  
Banach Spaces ◽  
Martingale Difference ◽  
Difference Sequences

A Generalization of Uniformly Rotund Banach Spaces

1979 ◽  
Vol 31 (3) ◽  
pp. 628-636 ◽  
Author(s):  
Francis Sullivan
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Boolean Algebra ◽  
Banach Spaces ◽  
Real Banach Space ◽  
Two Dimensional ◽  
Geometrical Characterization ◽  
Arbitrary Space ◽  
Image Position

Let X be a real Banach space. According to von Neumann's famous geometrical characterization X is a Hilbert space if and only if for all x, y ∈ XThus Hilbert space is distinguished among all real Banach spaces by a certain uniform behavior of the set of all two dimensional subspaces. A related characterization of real Lp spaces can be given in terms of uniform behavior of all two dimensional subspaces and a Boolean algebra of norm-1 projections [16]. For an arbitrary space X, one way of measuring the “uniformity” of the set of two dimensional subspaces is in terms of the real valued modulus of rotundity, i.e. for The space is said to be uniformly rotund if for each 0 we have .

scholarly journals Geometrical characterization of reflexivity in Banach spaces

1986 ◽  
Vol 30 ◽  
pp. 109-113
Author(s):  
M. J. Chasco ◽  
E. A. Indurain
Keyword(s):  
Banach Spaces ◽  
Geometrical Characterization

scholarly journals Fractional vector-valued Littlewood–Paley–Stein theory for semigroups

2014 ◽  
Vol 144 (3) ◽  
pp. 637-667 ◽  
Author(s):  
José L. Torrea ◽  
Chao Zhang
Keyword(s):  
Banach Space ◽  
Fractional Derivative ◽  
Banach Spaces ◽  
Martingale Difference ◽  
Area Function ◽  
Poisson Semigroup ◽  
Relationship Of ◽  
The Relationship ◽  
Vector Valued

We consider the fractional derivative of a general Poisson semigroup. With this fractional derivative, we define the generalized fractional Littlewood–Paley g-function for semigroups acting on Lp-spaces of functions with values in Banach spaces. We give a characterization of the classes of Banach spaces for which the fractional Littlewood–Paley g-function is bounded on Lp-spaces. We show that the class of Banach spaces is independent of the order of derivation and coincides with the classical (Lusin-type/-cotype) case. We also show that the same kind of results exist for the case of the fractional area function and the fractional gλ*-function on ℝn. Finally, we consider the relationship of the almost sure finiteness of the fractional Littlewood–Paley g-function, the area function and the gλ*-function with the Lusin-cotype property of the underlying Banach space. As a byproduct of the techniques developed, one can find some results of independent interest for vector-valued Calderón–Zygmund operators. For example, one can find the following characterization: a Banach space is the unconditional martingale difference if and only if, for some (or, equivalently, for every) p ∈ [1, ∞), dy exists for almost every x ∈ ℝ and every .

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