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

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