The Banach–Saks index of rearrangement invariant spaces on [0,1]

The set of all rearrangement invariant function spaces on [0,1] having the p-Banach–Saks property has a unique maximal element for all p∈(1,2]. For p=2 this is L₂, for p∈(1,2) this is L_p,∞⁰. We compute the Banach–Saks index for the families of Lorentz spaces $\text{L}{}_{\mathrm{p,q}}\text{,}\text{1<p<∞}$, 1?q?∞, and Lorentz–Zygmund spaces L(p,α), $\text{1?p<∞,}\text{α∈}\text{R}$, extending the classical results of Banach–Saks and Kadec–Pelczynski for L_p-spaces. Our results show that the set of rearrangement invariant spaces with Banach–Saks index p∈(1,2] is not stable with respect to the real and complex interpoltaion methods. To cite this article: E.M. Semenov, F.A. Sukochev, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Compact open sets in dual spaces and projections in group algebras of [FC]? groups

The structure of a compact open set in the dual of an [FC]^? group G, a locally compact group with relatively compact conjugacy classes, is given in terms of certain subsets which arise somewhat naturally. The support in the dual of a projection in L ¹(G) is a compact open set. Therefore, knowledge of the structure of such sets helps in identifying and constructing projections. We describe explicitly the compact open sets and construct projections for some illustrative examples.     

New characterizations of Hajlasz-Sobolev spaces on metric spaces

This paper introduces the fractional Sobolev spaces on spaces of homogeneous type,includingmetric spaces and fractals. These Sobolev spaces include the well-known Hajfasz-Sobolev spaces as specialmodels.The author establishes varions chaaracterizations of(sharp)maximal functions for these spaces.Asapplications,the author identifies the fractional Sobolev spaces with some Lipscitz-type spaces.Moreover;some embedding theorems are also given.     

The Szlenk index and local ℓ<Subscript>1</Subscript>-indices

We introduce two new local ₁-indices of the same type as the Bourgain ₁-index; the ⁺₁-index and the ⁺₁-weakly null index. We show that the ⁺₁-weakly null index of a Banach space X is the same as the Szlenk index of X, provided X does not contain ₁. The ⁺₁-weakly null index has the same form as the Bourgain ₁-index: if it is countable it must take values for some <₁. The different ₁-indices are closely related and so knowing the Szlenk index of a Banach space helps us calculate its ₁-index, via the ⁺₁-weakly null index. We show that I(C(^{))=^1++1}.     

Schur spaces and weighted spaces of type H ∞

Abstract

We extend some results related to composition operators on H _υ(G) to arbitrary linear operators on H _υ0(G) and H _υ(G). We also give examples of rank-one operators on H _υ(G) which cannot be approximated by composition operators.     

Square of ω-hyponormal operators

In this paper, we show that ifT is a -hyponormal operator, thenT ² is also -hyponormal.     

Embedding theorems of Besov and Triebel-Lizorkin spaces on spaces of homogeneous type

In this paper, the author establishes the embedding theorems for different metrics of inhomoge-neous Besov and Triebel-Lizorkin spaces on spaces of homogeneous type. As an application, the author obtains some estimates for the entropy numbers of the embeddings in the limiting cases between some Besov spaces and some logarithmic Lebesgue spaces.     

Isomorphisms of fiber spaces over Teichmüller spaces

We will be mainly concerned with some important fiber spaces over Teichmuller spaces, including the Bers fiber space and Teichmuller curve, establishing an isomorphism theorem between "punctured" Teichmuller curves and determining the biholomorphic isomorphisms of these fiber spaces.     

Compact composition operators between weighted Bergman spaces on convex domains in ℂ n

It is shown that a compact composition operator on a weighted Bergman space over a smoothly bounded strongly convex domain in ⁿ can have no angular derivative. Also, sufficient conditions for the boundedness and the compactness of composition operators defined on Hardy and weighted Bergman spaces are obtained, for situations in which each of the target spaces is enlarged in a natural way.     

New characterizations of Hajłasz-Sobolev spaces on metric spaces

This paper introduces the fractional Sobolev spaces on spaces of homogeneous type, including metric spaces and fractals. These Sobolev spaces include the well-known Hajłasz-Sobolev spaces as special models. The author establishes various characterizations of (sharp) maximal functions for these spaces. As applications, the author identifies the fractional Sobolev spaces with some Lipscitz-type spaces. Moreover, some embedding theorems are also given.     

On Compact Perturbations of m-Accretive Operators

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Linear isometries of Hardy spaces

According to results established by DeLeeuw-Rudin-Wermer and by Forelli,all linear isometries of any Hardy space H~p(p≥1,p≠2)on the open unit discΔof C are represented by weighted composition operators defined by inner functions onΔ.After reviewing(and completing when p=∞)some of those results,the present report deals with a characterization of periodic and almost periodic semigroups of linear isometries of H~p.     

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 ${\Delta }_{v,r}^m\left(c_0\right)$.     

An ω-hierarchy of axiom systemZF

In this paper we develop a sequenceZ ₀,...,Z _n ,...(n) of axiom systems for set theory, such that (1) the consistency of any system within the sequence is provable in its succeeding systems, (2) the first system in the sequence is Zermelo 's systemZ and the union of all systems in the sequence is justZF.     

Nice Enumerations of ω-categorical Groups

We give an example of an ω-categorical group without AZ-enumerations. We build AZ-enumerations of some ω-categorical central products of ω copies of a finite 2-step nilpotent group.     

Differentiable structure of ω-subsets of symplectic groups

In this paper, we study the differentiable structure of the ω-subset of Sp(2n), which is formed by all matrices in Sp(2n) possessing ω as an eigenvalue, for ω on the unit circle in the complex plane. Based on this result the ω -index theory parametrized by all ω on the unit circle for arbitrary symplectic paths is defined.     

ℂ-differentiability of mappings of Hilbert spaces

The process of differentiation is studied for a class of mappings of regions of a Hilbert space. We introduce the concept of Cl-differentiability of a mapping at a point and establish sufficient conditions of Cl-differentiability.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 3, pp. 424–428, March, 1995.     

Certain free products of operator spaces

Certain free products are introduced for operator spaces and dual operator spaces. It is shown that the free product of operator spaces does not preserve the injectivity. The linking C*-algebra of the full free product of two ternary rings of operators (simply, TRO's) is *-isomorphic to the full free product of the linking C*-algebras of the two TRO's. The operator space-reduced free product of the preduals of von Neumann algebras agrees with the predual of the reduced free product of the von Neumann algebras. Each of two operator spaces can be embedded completely isometrically into the reduced free product of the operator spaces. Finally, an example is presented to show that the C*-algebra-reduced free product of two C*-algebras may be contractively isomorphic to a proper subspace of their reduced free product as operator spaces.