Abstract It is known that not every Banach algebra has nontrivial
bounded derivations. For instance, consider large families of weighted
semisimple Banach algebras. In particular, we will be concerned with
derivations within the concrete frame of the nonabelian,
nonunitary, involutive Banach algebra $l^{2}(N^{2})$. The
theoretical interest in this algebra is based on the wellknown fact that it
is isomorphic to the class of HilbertSchmidt operators acting between two
given separable Hilbert spaces. In this article, we
characterize and determine the explicit structure of all bounded
derivations on $l^{2}(N^{2})$.
