Daugavet centers

An operator $G : \allowbreak X \to Y$ is said to be a Daugavet center if $\|G + T\| = \|G\| + \|T\|$ for every rank-1 operator $T : \allowbreak X \to Y$. The main result of the paper is: if $G : \allowbreak X \to Y$ is a Daugavet center, $Y$ is a subspace of a Banach space $E$, and $J: Y \to E$ is the natural embedding operator, then $E$ can be equivalently renormed in such a way, that $J \circ G : X \to E$ is also a Daugavet center. This result was previously known for particular case $X=Y$, $G=\mathrm{Id}$ and only in separable spaces. The proof of our generalization is based on an idea completely different from the original one. We give also some geometric characterizations of Daugavet centers, present a number of examples, and generalize (mostly in straightforward manner) to Daugavet centers some results known previously for spaces with the Daugavet property

Three Spaces Problem for Lyapunov Theorem on Vector Measure

It is proved that a Banach space X has the Lyapunov property if its subspace Y and the quotient space X/Y have it

Measurable selectors and set-valued Pettis integral in non-separable Banach spaces

AbstractKuratowski and Ryll-Nardzewski's theorem about the existence of measurable selectors for multi-functions is one of the keystones for the study of set-valued integration; one of the drawbacks of this result is that separability is always required for the range space. In this paper we study Pettis integrability for multi-functions and we obtain a Kuratowski and Ryll-Nardzewski's type selection theorem without the requirement of separability for the range space. Being more precise, we show that any Pettis integrable multi-function F:Ω→cwk(X) defined in a complete finite measure space (Ω,Σ,μ) with values in the family cwk(X) of all non-empty convex weakly compact subsets of a general (non-necessarily separable) Banach space X always admits Pettis integrable selectors and that, moreover, for each A∈Σ the Pettis integral ∫AFdμ coincides with the closure of the set of integrals over A of all Pettis integrable selectors of F. As a consequence we prove that if X is reflexive then every scalarly measurable multi-function F:Ω→cwk(X) admits scalarly measurable selectors; the latter is also proved when (X∗,w∗) is angelic and has density character at most ω1. In each of these two situations the Pettis integrability of a multi-function F:Ω→cwk(X) is equivalent to the uniform integrability of the family {supx∗(F(⋅)):x∗∈BX∗}⊂RΩ. Results about norm-Borel measurable selectors for multi-functions satisfying stronger measurability properties but without the classical requirement of the range Banach space being separable are also obtained

Classification of quantum groups and Belavin--Drinfeld cohomologies for orthogonal and symplectic Lie algebras

In this paper we continue to study Belavin-Drinfeld cohomology introduced in arXiv:1303.4046 [math.QA] and related to the classification of quantum groups whose quasi-classical limit is a given simple complex Lie algebra. Here we compute Belavin-Drinfeld cohomology for all non-skewsymmetric $r$-matrices from the Belavin-Drinfeld list for simple Lie algebras of type $B$, $C$, and $D$.Comment: 17 page

Dominated Convergence and Egorov Theorems for Filter Convergen

We study the filters, such that for convergence with respect to this filters the Lebesgue dominated convergence theorem and the Egorov theorem on almost uniform convergence are valid (the Lebesgue filters and the Egorov filters, respectively). Some characterizations of the Egorov and the Lebesgue filters are given. It is shown that the class of Egorov filters is a proper subset of the class of Lebesgue filters, in particular, statistical convergence filter is the Lebesgue but not the Egorov filter. It is also shown that there are no free Lebesgue ultrafilters. Significant attention is paid to the filters generated by a matrix summability method

Daugavet Centers

An operator G: X → Y is said to be a Daugavet center if ||G + T|| = ||G|| + ||T|| for every rank-1 operator T: X → Y . The main result of the paper is: if G: X →! Y is a Daugavet center, Y is a subspace of a Banach space E, and J : Y → E is the natural embedding operator, then E can be equivalently renormed in such a way that J ○ G : X → E is also a Daugavet center. This result was previously known for the particular case X = Y, G = Id and only in separable spaces. The proof of our generalization is based on an idea completely di®erent from the original one. We also give some geometric characterizations of the Daugavet centers, present a number of examples, and generalize (mostly in straightforward manner) to Daugavet centers some results known previously for spaces with the Daugavet property

Narrow Operators on Bochner L₁-Spaces

Narrow operators on L₁(μ;X)-spaces are studied. We present a sufficient criterion for such an operator to be narrow that resembles the characterization on narrow operators on L₁(μ) and show that the criterion is necessary for certain X, e.g., for spaces with the RNP