We investigate a new representation of general operators by means of sums of
shifted Gabor multipliers. These representations arise by studying the matrix
of an operator with respect to a Gabor frame. Each shifted Gabor multiplier
corresponds to a side-diagonal of this matrix. This representation is
especially useful for operators whose associated matrix possesses some
off-diagonal decay. In this case one can completely characterize the symbol
class of the operator by the size of the symbols of the Gabor multipliers. As
an application we derive approximation theorems for pseudodifferential
operators in the Sjostrand class.