New Majorization for Bounded Linear Operators in Hilbert Spaces

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