  The notation of generalized Bessel multipliers is obtained by a bounded
operator on $\ell^2$ which is inserted between the analysis and synthesis
operators. We show that various properties of generalized multipliers are
closely related to their parameters, in particular it will be shown that the
membership of generalized Bessel multiplier in the certain operator classes
requires that its symbol belongs in the same classes, in special sense. Also,
we give some examples to illustrate our results. As we shall see, generalized
multipliers associated with Riesz bases are well-behaved, more precisely in
this case multipliers can be easily composed and inverted. Special attention is
devoted to the study of invertible generalized multipliers. Sufficient and/or
necessary conditions for invertibility are determined. Finally, the behavior of
these operators under perturbations is discussed.
