In this paper, we focus on the duo ring property via quasinilpotent elements, which gives a new kind of generalizations commutativity. We call rings qnil-duo. Firstly, some properties quasinilpotents in are provided. Then set is applied to rings, perspective, introduce and study right (resp., left) qnil-duo rings. show that concept not left-right symmetric. Among others, it proved if Hurwitz se...

Let Sp be the Schatten-von Neumann ideal of compact operators equipped with the norm Np(·). For an A ∈ Sp (1 < p <∞), the inequality [ ∞ ∑ k=1 |Reλk(A)| ] 1 p + bp [ ∞ ∑ k=1 | Imλk(A)| ] 1 p ≥ Np(AR)− bpNp(AI) (bp = const. > 0) is derived, where λj(A) (j = 1, 2, . . . ) are the eigenvalues of A, AI = (A − A∗)/2i and AR = (A + A∗)/2. The suggested approach is based on some relations between the ...

Let X be a complex Hausdorff locally convex space. A system of seminorms P inducing the topology on X will be called a calibration. We denote by (X) the collection of all calibrations on X . For a given P ∈ (X) let P = {pα : α ∈ ∆}, where ∆ is some index set and for each α ∈ ∆ denote Uα = {x : pα(x) ≤ 1}. Let us denote by (X) the set of all linear continuous operators on X , by (X) the set of c...