Weak majorization and doubly substochastic operators on \(\ell^p(I)\)
Let \({\ell}^p(I)^+\) denote a set of all positive elements of the Banach space \({\ell}^p(I)\) with \(p\geq 1\), that is, \[ {\ell}^p(I)^+:=\{f\in{\ell}^p(I):f(i)\geq 0,\forall i\in I\}. \] Using the notion of a doubly substochastic operator, the author defines a weak majorization on \({\ell}^p(I)^+\) as follows. For \(f,g\in {\ell}^p(I)^+\), we say that \(f\) is weakly majorized by \(g\) if there exists a doubly substochastic operator \(D\) on \({\ell}^p(I)\) such that \(f=Dg\). Several properties of double substochastic operators and their relationship with the weak majorization are proved. In particular, it is shown that \(f\) is weakly majorized by \(g\) and \(g\) is weakly majorized by \(f\) if and only if there exists a partial permutation \(P\) such that \(g=Pf\).
row, column and doubly substochastic operators
scientific article; zbMATH DE number 6485763
scientific article