The S-basis and M-basis Problems for Separable Banach Spaces

This note has two objectives. The first objective is show that, even if a separable Banach space does not have a Schauder basis (S-basis), there always exists Hilbert spaces \mcH_1 and \mcH_2, such that \mcH_1 is a continuous dense embedding in \mcB and \mcB is a continuous dense embedding in \mcH_2. This is the best possible improvement of a theorem due to Mazur (see \cite{BA} and also \cite{PE1}). The second objective is show how \mcH_2 allows us to provide a positive answer to the Marcinkiewicz-basis (M-basis) problem

The M-basis Problem for Separable Banach Spaces

In this note we show that, if \mcB is separable Banach space, then there is a biorthogonal system $\{x_n, x_n^*\}$ such that, the closed linear span of \{x_n\},\bar{\left\langle {\{x_n\}}\right\rangle}=\mcB and $\left\| {x_n} \right\|\left\| {x_n^*} \right\| = 1$ for all $n$

Pelczynski's property (V) on spaces of vector valued functions

Let $E$ be a separable Banach space and $\Omega$ be a compact Hausdorff space. It is shown that the space $C(\Omega,E)$ has property (V) if and only if $E$ does. Similar result is also given for Bochner spaces $L^p(\mu,E)$ if $1<p<\infty$ and $\mu$ is a finite Borel measure on $\Omega$

An abstract result on Cohen strongly summing operators

We present an abstract result that characterizes the coincidence of certain classes of linear operators with the class of Cohen strongly summing linear operators. Our argument is extended to multilinear operators and, as a consequence, we establish a few alternative characterizations for the class of Cohen strongly summing multilinear operators.Comment: 9 page

A study of reciprocal Dunford-Pettis-like properties on Banach spaces

In this article, we study the relationship between $p$-$(V)$ subsets and p-$V^*$ subsets of dual spaces. We investigate the Banach space X with the property that adjoint every $p$-convergent operator $T: X \rightarrow Y$ is weakly $q$-compact, for every Banach space $Y$. Moreover, we define the notion of $q$-reciprocal Dunford-Pettis$\^*$property of order $p$ on Banach spaces and obtain a characterization of Banach spaces with this property. The stability of reciprocal Dunford-Pettis property of order $p$ for the projective tensor product is given

Fractional integrals and Fourier transforms

This paper gives a short survey of some basic results related to estimates of fractional integrals and Fourier transforms. It is closely adjoint to our previous survey papers \cite{K1998} and \cite{K2007}. The main methods used in the paper are based on nonincreasing rearrangements. We give alternative proofs of some results. We observe also that the paper represents the mini-course given by the author at Barcelona University in October, 2014.Comment: 42 page

On the distribution of Sidon series

Let B denote an arbitrary Banach space, G a compact abelian group with Haar measure $\mu$ and dual group $\Gamma$. Let E be a Sidon subset of $\Gamma$ with Sidon constant S(E). Let r_n denote the n-th Rademacher function on [0, 1]. We show that there is a constant c, depending only on S(E), such that, for all $\alpha > 0$: c^{-1}P[| \sum_{n=1}^Na_nr_n| >= c \alpha ] <= \mu[| \sum_{n=1}^Na_n\gamma_n| >= \alpha ] = c^{-1} \alpha

On pseudo weakly compact operators of order P P

In this paper, we introduce the concept of a pseudo weakly compact operator of order $p$ between Banach spaces. Also we study the notion of $p$-Dunford-Pettis relatively compact property which is in "general" weaker than the Dunford-Pettis relatively compact property and gives some characterizations of Banach spaces which have this property. Moreover, by using the notion of $p$-Right subsets of a dual Banach space, we study the concepts of $p$-sequentially Right and weak $p$-sequentially Right properties on Banach spaces. Furthermore, we obtain some suitable conditions on Banach spaces $X$ and $Y$ such that projective tensor and injective tensor products between $X$ and $Y$ have the $p$-sequentially Right property.\ Finally, we introduce two properties for the Banach spaces, namely $p$-sequentially Right$^{\ast}$ and weak $p$-sequentially Right$^{\ast}$ properties and obtain some characterizations of these properties

Positive operators as commutators of positive operators

It is known that a positive commutator $C=A B - B A$ between positive operators on a Banach lattice is quasinilpotent whenever at least one of $A$ and $B$ is compact. In this paper we study the question under which conditions a positive operator can be written as a commutator between positive operators. As a special case of our main result we obtain that positive compact operators on order continuous Banach lattices which admit order Pelczy\'nski decomposition are commutators between positive operators. Our main result is also applied in the setting of a separable infinite-dimensional Banach lattice $L^p(\mu)$ $(1<p<\infty)$.Comment: 20 page

On constructions of strong and uniformly minimal M-bases in Banach spaces

We find a natural class of transformations ("flattened perturbations") of a norming M-basis in a Banach space X, which give a strong norming M-basis in X. This simplifies and generalizes the positive answer to the "strong M-basis problem" solved by P. Terenzi. We also show that in general one cannot achieve uniformly minimality applying standard transformations to a given norming M-basis, despite of the existence in X a uniformly minimal strong M-bases.Comment: 10 page