in this article, we discuss measure theoretic characterizations for weighted composition operators in some operator classes on $l^{2}(sigma)$ such as, $n$-power normal, $n$-power quasi-normal, $k$-quasi-paranormal and quasi-class$a$. then we show that weighted composition operators can separate these classes.

The Brunn-Minkowski theory in convex geometry concerns, among other things, the volumes, mixed and surface area measures of bodies. We study generalizations these concepts to Borel with density Rn– particular, weighted versions volumes (the so-called measures) when dealing up three distinct then formulate analyze classical measures, obtain a new integral formula for measure As an application, w...

The weighted semigroup algebra Mb (S, w) is studied via its identification with Mb (S) together with a weighted algebra product *w so that (Mb (S, w), *) is isometrically isomorphic to (Mb (S), *w). This identification enables us to study the relation between regularity and amenability of Mb (S, w) and Mb (S), and improve some old results from discrete to general case.