On a generalized Semrl's theorem for weak-2-local derivations on B(H)

We prove that, for every complex Hilbert space $H$, every weak-2-local derivation on $B(H)$ or on $K(H)$ is a linear derivation. We also establish that every weak-2-local derivation on an atomic von Neumann algebra or on a compact C$^*$-algebra is a linear derivation.