  In this paper it is investigated how to find a matrix representation of
operators on a Hilbert space with Bessel sequences, frames and Riesz bases. In
many applications these sequences are often preferable to orthonormal bases
(ONBs). Therefore it is useful to extend the known method of matrix
representation by using these sequences instead of ONBs for these application
areas. We will give basic definitions of the functions connecting infinite
matrices defining bounded operators on l2 and operators on the Hilbert space.
We will show some structural results and give some examples. Furthermore in the
case of Riesz bases we prove that those functions are isomorphisms. Finally we
are going to apply this idea to the connection of Hilbert-Schmidt operators and
Frobenius matrices.
  Keywords: frames, discrete expansion, operators, matrix representation,
Hilbert-Schmidt operators, Frobenius matrices, Riesz bases.
