Generalized Frame Operator, Lower Semi-Frames and Sequences of Translates

Given an arbitrary sequence of elements xi={xi(n)}(n is an element of N) of a Hilbert space (H, ), the operator T-xi is defined as the operator associated to the sesquilinear form Omega xi(f,g)= Sigma(n is an element of N) < f,xi(n)>, for f,g is an element of{h is an element of H: Sigma(n is an element of N) vertical bar < h,xi n vertical bar(2)< infinity}. This operator is in general different from the classical frame operator but possesses some remarkable properties. For instance, T-xi is always self-adjoint with regard to a particular space, unconditionally defined, and, when xi is a lower semiframe, T-xi gives a simple expression of a dual of xi. The operator T-xi and lower semiframes are studied in the context of sequences of integer translates of a function of L-2(R). In particular, an explicit expression of T-xi is given in this context, and a characterization of sequences of integer translates, which are lower semiframes, is proved.