# Methods of Functional Analysis

and Topology

Editors-in-Chief: A. N. Kochubei,
Yu. G. Kondratiev

ISSN: 1029-3531 (Print) 2415-7503 (Online)

Founded by Yu. M. Berezansky in 1995.

Methods of Functional Analysis and Topology (MFAT), founded in 1995, is a peer-reviewed journal publishing original articles and surveys on general methods and techniques of functional analysis and topology with a special emphasis on applications to modern mathematical physics.

MFAT is an open access journal, free for authors and free for readers.

MFAT is indexed in: MathSciNet, zbMATH, Scopus, Web of Science, DOAJ, Google Scholar

## Latest Articles (September, 2019)

### Unbounded translation invariant operators on commutative hypergroups

Vishvesh Kumar, N. Shravan Kumar, Ritumoni Sarma

Methods Funct. Anal. Topology **25** (2019), no. 3, 236-247

Let $K$ be a commutative hypergroup. In this article, we study the unbounded translation invariant operators on $L^p(K),\, 1\leq p \leq \infty.$ For $p \in \{1,2\},$ we characterize translation invariant operators on $L^p(K)$ in terms of the Fourier transform. We prove an interpolation theorem for translation invariant operators on $L^p(K)$ and we also discuss the uniqueness of the closed extension of such an operator on $L^p(K)$. Finally, for $p \in \{1,2\},$ we prove that the space of all closed translation invariant operators on $L^p(K)$ forms a commutative algebra over the field of complex numbers. We also prove Wendel's theorem for densely defined closed linear operators on $L^1(K).$

### Limited and Dunford-Pettis operators on Banach lattices

Khalid Bouras, Abdennabi EL Aloui, Aziz Elbour

Methods Funct. Anal. Topology **25** (2019), no. 3, 205-210

This paper is devoted to investigation of conditions on a pair of Banach lattices $E; F$ under which every positive Dunford-Pettis operator $T:E\rightarrow F$ is limited. Mainly, it is proved that if every positive Dunford-Pettis operator $T:E\rightarrow F$ is limited, then the norm on $E'$ is order continuous or $F$ is finite dimensional. Also, it is proved that every positive Dunford-Pettis operator $T:E\rightarrow F$ is limited, if one of the following statements is valid:

1) The norm on $E^{\prime }$ is order continuous, and $F^{\prime }$ has weak$^{\ast }$ sequentially continuous lattice operations.

2) The topological dual $E^{\prime }$is discrete and its norm is
order continuous.

3) The norm of $E^{\prime }$ is order continuous and the lattice
operations in $E^{^{\prime }}$ are weak$^{\ast }$ sequentially continuous.

4) The norms of $E$ and of $E^{\prime }$ are order continuous.

### Space of configurations and the special measures on it

Methods Funct. Anal. Topology **25** (2019), no. 3, 197-204

The article is devoted to an exact account of initial results about configurations and measures on them, starting from the concept of a unique topologization of the space of all configurations, including both finite and infinite cases (not as it is made in the classical works).

### The Fourier transform on 2-step Lie groups

Methods Funct. Anal. Topology **25** (2019), no. 3, 248-272

In this paper, we study the Fourier transform on finite dimensional $2$-step Lie groups in terms of its canonical bilinear form (CBF) and its matrix coefficients. The parameter space of these matrix coefficients $\tilde{g}$, endowed with a distance $\rho_E$ which exchanges the regularity of a function with the decay of its Fourier matrix coefficients (cf. the Riemann-Lebesgue lemma for the classical Fourier transform) is however not complete. We compute explicitly its completion $\hat{g}$; the lack of completeness appears exactly when the CBF has nonmaximal rank. We provide an example for which partial degeneracy (partial rank loss) of the canonical form occurs, as opposed to the full degeneracy at the origin. We also compute the kernel $(w,\hat{w}) \mapsto \Theta(w,\hat{w})$ of the matrix-coefficients Fourier transform, the analogue for the $2$-step groups of the classical Fourier kernel $(x,\xi) \mapsto e^{i \langle \xi, x\rangle}$.