On innerness of derivations on $\mathcal{S(H)}$

We consider general bounded derivations on the Banach algebra of Hilbert–Schmidt operators on an underlying complex infinite dimensional separable Hilbert space $\mathcal H$. Their structure is described by means of unique infinite matrices. Certain classes of derivations are identified together in such a way that they correspond to a unique matrix derivation. In particular, Hadamard derivations, the action of general derivations on Hilbert–Schmidt and nuclear operators and questions about innerness are considered.