If ${\mathscr H}$ is a Hilbert space, $\varphi$ is a (not necessary linear) $*$-surjective mapping on ${\mathbb B}({\mathscr H})$ and $\varphi$ preserves the spectrum of operators of the form $ABA^{*}$, then $\varphi$ is either an algebra automorphism or an algebra anti-automorphism.

We present several operator versions of the Dunkl--Williams inequality with respect to the $p$-angular distance for operators. More precisely, we show that if $A, B \in \mathbb{B}(\mathscr{H})$ such that $|A|$ and $|B|$ are invertible, $\frac{1}{r}+\frac{1}{s}=1$ $(r>1)$ and $p\in\mathbb{R}$, then

\begin{equation*} \left|\,A|A|^{p-1}-B|B|^{p-1}\,\right|^{2} \leq |A|^{p-1}\left(\,r|A-B|^{2}+s\left|\,|A|^{1-p}|B|^{p}-|B|\,\right|^2\,\right)|A|^{p-1}.%\nonumber \end{equation*}

In the case that $0 < p \leq 1$, we remove the invertibility assumption and show that if $A=U|A|$ and $B=V|B|$ are the polar decompositions of $A$ and $B$, respectively, $t>0$, then

$$\left|\,\left(U|A|^{p}-V|B|^{p}\right)|A|^{1-p}\,\right|^{2}\leq \left(1+t\strut\right)|A-B|^{2}+\left(1+\frac{1}{t}\right) \left| |B|^{p}|A|^{1-p}-|B| \right|^2 .$$

We obtain several equivalent conditions, when the case of equalities hold.

We consider the stable rank and connected stable rank for certain non self-adjoint Banach algebras such as the triangular matrix algebras of all (finite or infinite) triangular matrices over a unital Banach algebra and certain nest algebras. Also, the stable rank estimate for certain crossed products of unital Banach algebras by isometries is obtained.

We first give a little improvement of nonlinear scalarizing function for vector optimization problem organized by Luc and Tammer-Weidner. As applications, we present Gordan's type alternative theorems for vector-valued function, optimality conditions for vector optimization problem and an existence theorem for vector saddle-points.

The purpose of the present paper is to establish necessary conditions and sufficient conditions that finite commutative hypergroups are extensions of finite Abelian groups by hypergroups of order two. Applying our results to some concrete cases one can determine all such extensions.