On Block diagonal majorization and Basic sequences

Abstract

‎In this paper we generalize (finite) block diagonal matrices to infinite dimensions and then by using block diagonal row stochastic matrices (as a special case)‎, ‎we define the relation $\prec_{_{bdr}}$ on $\mathfrak{c}_0,$ which is said block diagonal majorization‎.

‎We also obtain some important properties‎

‎of all bounded linear operators‎

‎$T:\mathfrak{c}_{0}\rightarrow\mathfrak{c}_{0},$‎

‎which preserve $\prec_{_{bdr}}.$‎

‎Further‎, ‎it is obtained necessary conditions for $T\in\mathcal{P}_{bdr},$ the set of all bounded linear operators on‎

‎$\mathfrak{c}_{0},$ which preserve block diagonal majorization‎.

‎Also‎, ‎the notion of the basic sequences correspond to block diagonal row stochastic‎

‎matrices with description of some relevant examples will be discussed.