Journal of Linear and Topological Algebra
Volume:12 Issue: 3, Summer 2023

In this paper, we introduce a new strong form of the continuity of multifunctions with the help of minimal open sets. We give some characterizations for this new continuity and investigate fundamental properties of it. Additionally, we use this type of multifunctions to characterize Alexandroff spaces.

‎Let $\mathsf{G}$ be a discrete group acting on $C^*$-algebra $\Im$‎. ‎In this paper‎, ‎we investigate projectivity and injectivity of $G$-Hilbert $\Im$-modules and study the equivalent conditions characterizing $\mathsf{G}$-$C^*$-subalgebras of the algebra of compact operators on $\mathsf{G}$-Hilbert spaces in terms of general properties of $\mathsf{G}$-Hilbert $\Im$-modules‎. ‎In particular‎, ‎we show that $\mathsf{G}$-Hilbert $\Im$-(bi)modules on $\mathsf{G}$-$C^*$-algebra of compact operators are both projective and injective‎.

The main aim of the present work is to review and study a variational method in existence and multiplicity of positive solutions for quasilinear elliptic systems with critical Hardy-Sobolev and sign-changing function exponents.

‎Let $ \mathcal{A} $ be a unital Banach algebra‎, ‎$ w\in \mathcal{A}$‎, ‎and $ \gamma‎ : ‎\mathcal{A} \to \mathcal{A} $ is a continuous linear map‎. ‎We show that $\gamma$ satisfies $a\gamma(b)=\gamma(w)$ ($\gamma(a)b=\gamma(w)$) whenever $a,b\in \mathcal{A}$ with $ab=w$ and $w$ is a left (right) separating point in $\mathcal{A}$ if and only if $\gamma$ is a right (left) centralizer‎. ‎Also‎, ‎we prove that $\gamma$ satisfies $a\gamma(b)=\gamma(a)b=\gamma(w)$ whenever $a,b\in \mathcal{A}$ with $ab=w$ and $w$ is a left or right separating point in $\mathcal{A}$ if and only if $\gamma$ is a centralizer‎. ‎We also provide some applications of the obtained results for characterization of a continuous linear map $\gamma:\mathcal{A}\rightarrow \mathcal{A}$ on a unital Banach $*$-algebra $\mathcal{A}$ satisfying $a\gamma(b)^{*}=\gamma(w^{*})^{*}$ ($\gamma(a)^{*}b=\gamma(w^{*})^{*}$) whenever $a,b\in \mathcal{A}$ with $ab^{*}=w$ ($a^{*}b=w$) and $w$ is a left (right) separating point‎, ‎or $\gamma$ satisfying $a\gamma(b)^{*}=\gamma(c)^{*}d=\gamma(w^{*})^{*}$ whenever $a,b,c,d\in \mathcal{A}$ with $ab^{*}=c^{*}d =w$ and $w$ is a left or right separating point‎.

‎The aim of this paper is to introduce the concept of multi-valued contraction that combine a renowned \'{C}iri\'{c}-type contraction and Caristi-type contractions in the framework of metric spaces‎. ‎The existence of fixed points for such contractions equipped with some suitable hypotheses are proved and some analogues of the fixed point theorems presented herein are deduced as corollaries‎. ‎Moreover‎, ‎an example is given to illustrate the validity of obtained main result‎.

The notion of compression has received enormous attention in recent years because of its necessity in terms of the computational cost and other applicable features. But many times the notion expansion appears to be quite useful. Tight frames are quite useful in signal reconstruction, signal and image de-noising, compressed sensing because of the availability of a simple, explicit reconstruction formula. So in this paper, we discuss the extension of a basis by including some very sparse (at most two nonzero components) vectors so that the new frame becomes a tight frame. We do the basis extension in finite dimensional Hilbert spaces (both real and complex) to construct tight frames. We formulate constructive algorithms to do the aforementioned task. The algorithms guarantee us to produce tight frames with very less computational cost, and the new tight frames compensate for multiple erasures. The algorithms also do not disturb the vectors in the given basis. We also present one application of the aforementioned concept.‎‎

Let G be a graph with each vertex is colored either white or black. A white vertex is changed to a black vertex when it is the only white neighbor of a black vertex (color-change rule). A zero forcing set S of a graph G is a subset of vertices G with black vertices, all other vertices G are white, such that after finitely many applications of the color-change rule all of vertices G becomes black. The zero forcing number of G is the minimum cardinality of a zero forcing set in G, denoted by Z(G). In this paper, we define ℓ−Path graphs. We give some ℓ−Path and ℓ−Ciclo graphs such that their maximum nullity are equal to their zero forcing number. Also, we obtain minimum propagation time and maximum propagation time for them.