Weak martingale Hardy spaces and weak atomic decompositions

In this paper we define some weak martingale Hardy spaces and three kinds of weak atoms. They are the counterparts of martingale Hardy spaces and atoms in the classical martingale Hp-theory. And then three atomic decomposition theorems for martingales in weak martingale Hardy spaces are proved. With the help of the weak atomic decompositions of martingale, a sufficient condition for a sublinear operator defined on the weak martingale Hardy spaces to be bounded is given. Using the sufficient condition, we obtain a series of martingale inequalities with respect to the weak Lp-norm, the inequalities of weak (p ,p)-type and some continuous imbedding relationships between various weak martingale Hardy spaces. These inequalities are the weak versions of the basic inequalities in the classical martingale Hp-theory.     

Atomic decompositions of martingale hardy–morrey spaces

We establish the atomic decompositions of martingale Hardy– Morrey spaces. The martingale Hardy–Morrey spaces are generalizations of martingale Hardy spaces. Therefore, the atomic decompositions presented in this paper are extensions of the atomic decompositions of martingale Hardy spaces.     

A construction of ℒ∞-spaces and related Banach spacesand related Banach spaces

Let λ>1. We prove that every separable Banach space E can be embedded isometrically into a separable ℒ _∞ ^λ -spaceX such thatX/E has the RNP and the Schur property. This generalizes a result in [2]. Various choices ofE allow us to answer several questions raised in the literature. In particular, takingE = ℓ₂, we obtain a ℒ _∞ ^λ -spaceX with the RNP such that the projective tensor product containsc ₀ and hence fails the RNP. TakingE=L ¹, we obtain a ℒ _∞ ^λ -space failing the RNP but nevertheless not containingc ₀.     

Algebraic structures of Makarevič spaces. I

We generalize features of bounded symmetric domains to a bigger class of symmetric spaces calledMakarevi spaces: we associate a generalizedBergman operator to such a space and describe the invariant pseudo-metric and the invariant measure on the space by means of this family of operators. The space itself can be characterized essentially as the domain where the generalized Bergman operator is nondegenerate. These results are applied to the theory ofcompactly causal symmetric spaces: we describe explicitly the complex domain associated to such a space.     

Approximative compactness and continuity of metric projector in Banach spaces and applications

First we prove that the approximative compactness of a nonempty set C in a normed linear space can be reformulated equivalently in another way.It is known that if C is a semi-Chebyshev closed and approximately compact set in a Banach space X,then the metric projectorπC from X onto C is continuous.Under the assumption that X is midpoint locally uniformly rotund,we prove that the approximative compactness of C is also necessary for the continuity of the projectorπC by the method of geometry of Banach spaces.Using this general result we find some necessary and sufficient conditions for T to have a continuous Moore-Penrose metric generalized inverse T~ ,where T is a bounded linear operator from an approximative compact and a rotund Banach space X into a midpoint locally uniformly rotund Banach space Y.     

Geometry of Banach spaces with property β

We prove that every Banach space can be isometrically and 1-complementably embedded into a Banach space which satisfies property β and has the same character of density. Then we show that, nevertheless, property β satisfies a hereditary property. We study strong subdifferentiability of norms with property β to characterize separable polyhedral Banach spaces as those admitting a strongly subdifferentiable β norm. In general, a Banach space with such a norm is polyhedral. Finally, we provide examples of non-reflexive spaces whose usual norm fails property β and yet it can be approximated by norms with this property, namely (L ₁[0,1], ‖·‖1) and (C(K), ‖·‖∗) whereK is a separable Hausdorff compact space To the memory of A. Plans Supported in part by DGICYT grant PB 94-0243 and DGICYT PB 96-0607.     

On a Volume Parameter of Banach spaces

Having discussed the parameter in [5] we introduce in this paper a volume parameter We give some properties of it, and obtain that P_3(x)<4 if X is 2UR, and that X is a super-reflexive Banach space if p_3(x)<4.     

Some G-M-type Banach spaces and K-groups of operator algebras on them

By providing several new varieties of G-M-type Banach spaces according todecomposable and compoundable properties, this paper discusses the operator structuresof these spaces and the K-theory of the algebra of the operators on these G-M-type Ba-nach spaces through calculation of the K-groups of the operator ideals contained in theclass of Riesz operators.     

On Isomorphisms of Marušič–Scapellato Graphs

Maru?i?–Scapellato graphs are vertex-transitive graphs of order \(m(2^k + 1)\), where m divides \(2^k - 1\), whose automorphism group contains an imprimitive subgroup that is a quasiprimitive representation of \(\mathrm{SL}(2,2^k)\) of degree \(m(2^k + 1)\). We show that any two Maru?i?–Scapellato graphs of order pq, where p is a Fermat prime, and q is a prime divisor of \(p - 2\), are isomorphic if and only if they are isomorphic by a natural isomorphism derived from an automorphism of \(\mathrm{SL}(2,2^k)\). This work is a contribution towards the full characterization of vertex-transitive graphs of order a product of two distinct primes.     

Atomic decompositions and Hardy’s inequality on weak Hardy-Morrey spaces

We introduce the weak Hardy-Morrey spaces in this paper. We also obtain the atomic decompositions of the weak Hardy-Morrey spaces. By using these decompositions, we establish the Hardy inequalities on the weak Hardy-Morrey spaces.     

On the ℂ-differentiability of mappings of Banach spaces

We study the monogeneity conditions for -differentiable mappings of domains of a Banach space and establish criteria of -differentiability of a mapping at a point.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 10, pp. 1336–1342, October, 1994.     

Existence and approximation of fixed points of multivalued generalized α-nonexpansive mappings in Banach spaces

Numerical Algorithms - We consider the theoretical and numerical aspects of the quadrature rules associated with a sequence of polynomials generated by a special RII recurrence relation. We also...     

Polar decompositions and related classes of operators in spaces ∏κ

Polar decompositions with respect to an indefinite inner product are studied for bounded linear operators acting on a space. Criteria are given for existence of various forms of the polar decompositions, under the conditions that the range of a given operatorX is closed and that zero is not an irregular critical point of the selfadjoint operatorX ^[*]X. Both real and complex spaces are considered. Relevant classes of operators having a selfadjoint (in the sense of the indefinite inner product) square root, or a selfadjoint logarithm, are characterized.The work of this author was partially supported by INdAM-GNCS and MURSTThe work of this author was partially supported by NSF grant DMS-9988579.     

Analog of Borsuk’s problem on Banach spaces

An analog of the well-known problem of partition of figures into parts with less diameter in Banach spaces is studied. The sufficient conditions for the sets to belong to the class of Borsuk’s sets in multidimensional Banach spaces are first obtained.     

More on stability of almost surjective ε-isometries of Banach spaces

Let X and Y be two Banach spaces,and f:X→Y be a standard ε-isometry for some ε = 0.In this paper,by using a recent theorem established by Cheng et al.(2013–2015),we show a sufficient condition guaranteeing the following sharp stability inequality of f:There is a surjective linear operator T:Y→X of norm one so that ||T f(x)-x||= 2ε,for all x∈X.As its application,we prove the following statements are equivalent for a standard ε-isometry f:X→Y:(i)lim inf_(t→∞) dist(ty,f(X))/|t|1/2,for all y∈S_Y;(ii)τ(f)≡sup_(y∈S_Y) lim inf_(t→∞) dist(ty,f(X))/|t|=0;(iii)there is a surjective linear isometry U:X→Y so that || f(x)-Ux||= 2ε,for all x∈X.This gives an affirmative answer to a question proposed by Vestfrid(2004,2015).     

Gâteaux differentiability and uniform monotone approximation of convex functions in Banach spaces

Acta Mathematica Hungarica - We prove that if $$X^{*}$$ is strictly convex, a convex function $$f$$ is coercive and b-Lipschitzian iff there exist two convex function sequences...