Abstract

In this paper, we study the boundedness of weighted composition operators between two $L^p$-spaces. We present equivalent conditions for the compactness of weighted composition operators between two $L^p$-spaces. We also give equivalent conditions for weighted composition operators with closed range.

Keywords: Closed range; compact operator; measurable transformation; weighted composition operator.

MSC: 47B33, 46E30, 47B07, 46B70

Abstract

We define an equitorsion conform mapping of two generalized Riemannian spaces and obtain some invariant geometric objects of this mapping, generalizing the tensor of conform curvature.

Keywords: Conform mapping; generalized Riemannian space; equitorsion conform mapping; equitorsion conform curvature tensor.

MSC: 53B05

Abstract

We introduce the notion of double $cs$-cover and give a characterization on sequence-covering $\pi$-$s$-images of locally separable metric spaces by means of double $cs$-covers having $\pi$-property of $\aleph_0$-spaces.

Keywords: Sequence-covering, double $cs$-cover, point-star network, $\pi$-property, $\pi$-mapping, $s$-mapping.

MSC: 54D65, 54E35, 54E40

Abstract

We introduce new supplementary data to the set of eigenvalues, to determine uniquely the potential and boundary conditions of the Sturm-Liouville problem. As a corollary we obtain extensions of some known uniqueness theorems in the inverse Sturm-Liouville problem.

Keywords: Inverse Sturm-Liouville problem; uniqueness theorem; spectral data.

MSC: 34L20, 47E05

Abstract

A space $X$ is cs-starcompact if for every open cover $\Cal U$ of $X$, there exists a convergent sequence $S$ of $X$ such that $St(S,\Cal U)=X$, where $St(S,{\Cal U})=\bigcup\{U\in{\Cal U}:U\cap S\neq\emptyset\}$. In this note, we investigate the closed subspaces of cs-starcompact spaces.

Keywords: Countable compact; cs-starcompact.

MSC: 54D20, 54G20

Abstract

In this paper we will introduce the notion of $a$-connected elements of a semigroup, $a$-connected semigroups, and weakly externally commutative semigroup, and we prove that a weakly externally commutative semigroup is a semilattice of $a$-connected semigroups. Undefined notions can be found in [4].

Keywords: Semigroups; semilattice; Archimedean semigroups; externally commutative semigroup.

MSC: 20M10

Abstract

The notion of generalized 2-normed spaces was introduced by Lewandowska in 1999~[5]. One can obtain a generalized 2-normed space from a normed space. We shall define the notions of 1-type $\varepsilon$-quasi Chebyshev subspaces and give some results in this field.

Keywords: Generalized 2-normed spaces; $y$-subadditive map; 1-type $\varepsilon$-quasi Chebyshev subspace.

MSC: 46A15, 41A65

Abstract

Using fixed point theory, B. Brosowski [2] proved that if $T$ is a nonexpansive linear operator on a normed linear space $X$, $C$ a $T$-invariant subset of $X$ and $x$ a $T$-invariant point, then the set $P_C(x)$ of best $C$-approximant to $x$ contains a $T$-invariant point if $P_C(x)$ is non-empty, compact and convex. Subsequently, many generalizations of the Brosowski's result have appeared. We also obtain some results on invariant points of a nonexpansive mapping for the set of $\varepsilon$-approximation in metric spaces thereby generalizing and extending some known results including that of Brosowski, on the subject.

Keywords: $\varepsilon$-approximation; $\varepsilon$-coapproximation; convex metric space; $G$-convex structure; convex set; starshaped set; nonexpansive map and contraction map.

MSC: 41A50, 41A65, 47H10, 54H25

Abstract

In this paper we consider the notion of almost Menger property which is similar to the familiar property of Menger and prove that we can use regular open sets instead of open sets in the definition of almost Menger property. We give conditions for a space $X^n$ to be almost Menger. In the similar way, we consider almost $\gamma$-sets and the almost star-Menger property.

Keywords: Menger space; almost Menger space; star-Menger space; almost $\gamma$-set.

MSC: 54D20