Divisibility in the K0‐Group of the Algebra of Operators on a Banach Space

Let Figiel's reflexive Banach space which is not isomorphic to its Cartesian square. We show that the K ₀group of the algebra of continuous, linear operators on contain a subgroup isomorphic to the group c ₀₀( ) of sequences rational numbers with z _n=0 eventually.     

Fourier multipliers and spectral measures in Banach function spaces

It is classical that amongst all spaces L^p (G), 1 ≤ p ≤ ∞, for , or say, only L² (G) (that is, p = 2) has the property that every bounded Borel function on the dual group Γ determines a bounded Fourier multiplier operator in L² (G). Stone’s theorem asserts that there exists a regular, projection-valued measure (of operators on L² (G)), defined on the Borel sets of Γ, with Fourier-Stieltjes transform equal to the group of translation operators on L² (G); this fails for every p ≠ 2. We show that this special status of L² (G) amongst the spaces L^p (G), 1 ≤ p ≤ ∞, is actually more widespread; it continues to hold in a much larger class of Banach function spaces defined over G (relative to Haar measure).      

Banach 空间上算子的几种不可约性

本文根据Cowen-Douglas 算子的定义引入两类与强不可约算子有紧密联系的算子类— B_n 算子与B 算子. 说明了在可分Banach 空间上存在B_n 算子与B 算子. 文章详细讨论了几种具有不可约性的算子类之间的关系并得到了一个关系图. 本文还给出这些算子类的一些性质, 包括算子的(拟) 相似不变性等.     

K-Theory for the Banach Algebra of Operators on James''s Quasi-reflexive Banach Spaces

We prove that the K-groups of the Banach algebra of bounded, linear operators on the pth James space , where 1 < p < , are given by and . Moreover, for each Banach space and each non-zero, closed ideal contained in the ideal of inessential operators, we show that and . This enables us to calculate the K-groups of for each Banach space which is a direct sum of finitely many James spaces and -spaces.     

On Uniformly Finitely Extensible Banach spaces

We continue the study of Uniformly Finitely Extensible Banach spaces (in short, UFO) initiated in Moreno and Plichko (2009) [39] and Castillo and Plichko (2010) [18]. We show that they have the Uniform Approximation Property of Pe?czyński and Rosenthal and are compactly extensible. We will also consider their connection with the automorphic space problem of Lindenstrauss and Rosenthal – do there exist automorphic spaces other than _c0(I)$c_0(I)$ and _?2(I)${?}_2(I)$? – showing that a space all whose subspaces are UFO must be automorphic when it is Hereditarily Indecomposable (HI), and a Hilbert space when it is either locally minimal or isomorphic to its square. We will finally show that most HI – among them, the super-reflexive HI space constructed by Ferenczi – and asymptotically _?2${?}_2$ spaces in the literature cannot be automorphic.     

A Family of Enlargements of Maximal Monotone Operators

We introduce a family of enlargements of maximal monotone operators. The Brønsted and Rockafellar -subdifferential operator can be regarded as an enlargement of the subdifferential. The family of enlargements introduced in this paper generalizes the Brønsted and Rockafellar -subdifferential (enlargement) and also generalize the enlargement of an arbitrary maximal monotone operator recently proposed by Burachik, Iusem and Svaiter. We characterize the biggest and the smallest enlargement belonging to this family and discuss some general properties of its members. A subfamily is also studied, namely the subfamily of those enlargements which are also additive. Members of this subfamily are formally closer to the -subdifferential. Existence of maximal elements is proved. In the case of the subdifferential, we prove that the -subdifferential is maximal in this subfamily.     

A note on nice operators

In this note, we characterize nice operators in a class of Banach spaces, which includes spaces and L₁(μ), as those operators that preserve extreme points.     

Distortion of Banach spaces and supermultiplicative operational quantities

We prove that certain operational quantities q which characterize upper-semi Fredholm operators are supermultiplicative, in the sense of that q(S)q(T)?q(ST). Based on the distortion of Banach spaces we show that another is not supermultiplicative. Moreover we introduce two supermultiplicative operational quantities which characterize also the upper-semi Fredholm operators and we prove that they are not equivalent to some operational quantities known.     

Factorizing operators on Banach function spaces through spaces of multiplication operators

In order to extend the theory of optimal domains for continuous operators on a Banach function space X(μ) over a finite measure μ, we consider operators T satisfying other type of inequalities than the one given by the continuity which occur in several well-known factorization theorems (for instance, Pisier Factorization Theorem through Lorentz spaces, pth-power factorable operators …). We prove that such a T factorizes through a space of multiplication operators which can be understood in a certain sense as the optimal domain for T. Our extended optimal domain technique does not need necessarily the equivalence between μ and the measure defined by the operator T and, by using δ-rings, μ is allowed to be infinite. Classical and new examples and applications of our results are also given, including some new results on the Hardy operator and a factorization theorem through Hilbert spaces.     

非线性Lipschitz连续算子的定量性质（Ⅳ）──谱理论

本文将有界线性算子谱的定义及若干重要谱性质推广至非线性Ｌｉｐｓｃｈｉｔｚ连续算子,并得到一些有意义的结果．