Banach frames for multivariate α-modulation spaces

The α-modulation spaces , α∈[0,1], form a family of spaces that include the Besov and modulation spaces as special cases. This paper is concerned with construction of Banach frames for α-modulation spaces in the multivariate setting. The frames constructed are unions of independent Riesz sequences based on tensor products of univariate brushlet functions, which simplifies the analysis of the full frame. We show that the multivariate α-modulation spaces can be completely characterized by the Banach frames constructed.     

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.     

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.     

A Banach–Zarecki Theorem for functions with values in Banach spaces

A Banach–Zarecki Theorem for a Banach space-valued function  \(F : [0,1] \rightarrow X\) with compact range is presented. We define the strong absolute continuity ( \(sAC_{||.||_{F}}\) ) and the bounded variation ( \(BV_{||.||_{F}}\) ) of \(F\) with respect to the Minkowski functional \(||.||_{F}\) associated to the closed absolutely convex hull \(C_{F}\) of \(F([0,1])\) . It is proved that \(F\) is \(sAC_{||.||_{F}}\) if and only if \(F\) is \(BV_{||.||_{F}}\) , weak continuous on \([0,1]\) and satisfies the weak property \((N)\) .     

On pα-dual spaces of generalized difference sequence spaces

In this paper, we compute the $p\alpha$-, $p\beta$- and $p\gamma$-duals of the sequence spaces ${?}_{\infty }\left(k^m\right),c\left(k^m\right),c_0\left(k^m\right)$, the $pN$-dual of the sequence spaces ${\Delta }_{v,r}^m\left({?}_{\infty }\right),{\Delta }_{v,r}^m\left(c\right)$ and the $p\alpha$-dual of the sequence spaces ${\Delta }_{v,r}^m\left({?}_{\infty }\right),{\Delta }_{v,r}^m\left(c\right)$ and ${\Delta }_{v,r}^m\left(c_0\right)$.     

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.     

Embedding ofl ∞ k in finite dimensional Banach spaces

Letx ₁,x ₂, ...,x _n ben unit vectors in a normed spaceX and defineM _n =Ave{‖Σ _i=1 ⁿ ε₁ x _i ‖:ε₁=±1}. We prove that there exists a setA⊂{1, ...,n} of cardinality such that {x _i } _i∈A is 16M _n -isomorphic to the natural basis ofl _∞ ^k . This result implies a significant improvement of the known results concerning embedding ofl _∞ ^k in finite dimensional Banach spaces. We also prove that for every ∈>0 there exists a constantC(∈) such that every normed spaceX _n of dimensionn either contains a (1+∈)-isomorphic copy ofl ₂ ^m for somem satisfying ln lnm≧1/2 ln lnn or contains a (1+∈)-isomorphic copy ofl _∞ ^k for somek satisfying ln lnk>1/2 ln lnn−C(∈). These results follow from some combinatorial properties of vectors with ±1 entries. The contribution of the first author to this paper forms part of his Ph.D. Thesis written under the supervision of Prof. M. A. Perles from the Hebrew University.     

On quasimöbius maps in real Banach spaces

Suppose that E and E′ denote real Banach spaces with dimension at least 2, that D $ \subseteq $ E and D′ $ \subseteq $ E′ are domains, that f: D → D′ is an (M,C)-CQH homeomorphism, and that D is uniform. The aim of this paper is to prove that D′ is a uniform domain if and only if f extends to a homeomorphism $\overline f :\overline D \to {\overline D ^\prime }$ and $\overline f $ is η-QM relative to ?D. This result shows that the answer to one of the open problems raised by Väisälä from 1991 is affirmative.     

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.     

The structure of uniformly Gâteaux smooth Banach spaces

It is shown that a Banach spaceX admits an equivalent uniformly Gateaux smooth norm if and only if the dual ball ofX* in its weak star topology is a uniform Eberlein compact. Supported by AV 101-97-02, AV 1019003 and GA ČR 201-98-1449. Supported by GA ČR 201-98-1449, AV 1019003 and GAUK 1/1998.     

On (N,ϵ)-pseudospectra of operators on Banach spaces

In this paper we extend the concept of the $(N,?)$-pseudospectra of Hansen to the case of bounded linear operators on Banach spaces and prove several relations to the usual spectrum. We particularly discuss the approximation by rectangular finite sections and the impact of the fundamental result of Shargorodsky on “jumping” pseudospectra.     

Markuševič decompositions and quasireflexivity of Banach spaces

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On K_0-groups of operator algebras on Banach spaces

This paper merges some classifications of G-M-type Banach spaces simplifically, discusses the condition of K0(B(X)) = 0 for operator algebra B(X) on a Banach space X, and obtains a result to improve Laustsen's sufficient condition, gives an example to show that X ≈ X2 is not a sufficient condition of K0(B(X)) = 0.     

Weakly relaxed α-pseudomonotonicity and equilibrium problem in Banach spaces

In this paper, we introduce the weakly relaxed ??-pseudomonotone mapping which generalizes the class of pseudomonotone mappings. Some examples are given which show that the weakly relaxed ??-pseudomonotonicity is a proper generalization of pseudomonotonicity for both scalar and vector valued bi-functions. Using the KKM technique, we establish the existence of solutions of equilibrium problems and vector equilibrium problems with weakly relaxed ??-pseudomonotonicity in the reflexive Banach spaces. The present work extends some corresponding results of the variational-like inequalities (Fang and Huang in J. Optim. Theory Appl. 118:327?C338, 2003; Lee and Lee in J. Korea Soc. Math. Educ. Ser. B Pure Appl. Math. 11: 231?C242, 2004) to equilibrium problems.