NEW MAJORIZATION FOR BOUNDED LINEAR OPERATORS IN HILBERT SPACES

‎This work aims to introduce and investigate a preordering in $B(\mathcal{H}),$‎ ‎the Banach space of all bounded linear operators defined on a complex‎ ‎Hilbert space $\mathcal{H}.$ It is called strong majorization and denoted by $S\prec_{s}T,$ for‎ ‎$S,T\in B(\mathcal{H}).$ The strong majorization follows majorization defined by Barnes‎, ‎but not vice versa‎. ‎If $S\prec_{s}T,$ then $S$ inherits some properties of $T.$ ‎‎‎‎ The strong majorization will be extended for the d-tuple of operators in $B(\mathcal{H})^{d}$ and‎ ‎is called joint strong majorization denoted by $S\prec_{js}T,$ for $S,T\in B(\mathcal{H})^{d}.$ We show that‎ ‎some properties of strong majorization are satisfied for joint strong majorization‎.