Linear rank preservers on infinite triangular matrices

We consider T∞(F) - the space of all in nite upper triangular matrices over a eld F. We give a description of all linear maps that satisfy the property: if rank(x) = 1, then rank(φ(x)) = 1 for all x ∈ T∞(F). Moreover, we characterize all injective linear maps on T∞(F) such that if rank(x) = k, then rank(φ(x)) = k.