  In this paper we deal with the connection of frames with the class of Hilbert
Schmidt operators. First we give an easy criteria for operators being in this
class using frames. It is the equivalent to the criteria using orthonormal
bases. Then we construct Bessel sequences frames and Riesz bases for the class
of Hilbert Schmidt operators using the tensor product of such sequences in the
original Hilbert space. We investigate how the Hilbert Schmidt inner product of
an arbitrary operator and such a rank one operator can be calculated in an
efficient way. Finally we give an algorithm to find the best approximation, in
the Hilbert Schmidt sense, of arbitrary matrices by operators called frame
multipliers, which multiply the analysis coefficents by a fixed symbol followed
by a synthesis.
