On the duality of function valued frames in L^۲(۰,\infty)

By the notion of F_a-factorable operators, we establish a new version of the Riesz Representation Theorem adapted to this class of operators. Also, for F_a-factorable operators T and T^{\prime}, we derive some equivalent conditions such that the equation T^{\prime}=TX has a unique solution in the class of F_a-factorable operators. Moreover, by using the concept of F_a-frame in L^۲(۰,\infty), which is similar yet distinct from the traditional frame concept, we establish optimal F_a-frame bounds, and provide a complete characterization of all corresponding F_a-dual frames.

کلیدواژه ها:

F_a-inner product ، F_a-frame ، F_a-dual frame ، F_a-factorable\-function ، Riesz Representation Theorem