c-frames and c-bessel mappings

the theory of c-frames and c-bessel mappings are the generalizationsof the theory of frames and bessel sequences. in this paper, weobtain several equivalent conditions for dual of c-bessel mappings.we show that for a c-bessel mapping $f$, a retrievalformula with respect to a c-bessel mapping $g$ is satisfied if andonly if $g$ is sum of the canonical dual of $f$ with a c-besselmapping which  weakly belongs to the null space of the pre-frame operatorof $f$. also, we prove that composition of pre-frame operator withanalysis operator of two square norm integrable c-bessel mappingsare trace class operators.