A new class of linear operators on l2 and Schur multipliers for them

We introduce the space B-w(l(2)) of linear (unbounded) operators on l(2) which map decreasing sequences from l(2) into sequences from l(2) and we find some classes of operators belonging either to B-w(l(2)) or to the space of all Schur multipliers on B-w(l(2)). For instance we show that the space B(l(2)) of all bounded operators on l(2) is contained in the space of all Schur multipliers on B-w(l(2)).