We introduce the spaceBw(ℓ2)of linear (unbounded) operators onℓ2which map decreasing sequences fromℓ2into sequences fromℓ2and we find some classes of operators belonging either toBw(ℓ2)or to the space of all Schur multipliers onBw(ℓ2). For instance we show that the spaceB(ℓ2)of all bounded operators onℓ2is contained in the space of all Schur multipliers onBw(ℓ2).

We consider the infinite matrix version B(D, ℓ2) of the Bloch space. In this paper we complement the results in the recent book [18] by deriving some new characterizations of B(D, ℓ2) and related spaces and duals via Schur multipliers. As applications we find the largest solid subspace of B(D, ℓ2) and its "conjugate" space I(ℓ2) considered in [18] and [19].