Generalized frame operator, lower semi-frames and sequences of translates

Given a sequence of elements $xi={xi_n}_{nin mathbb{N}}$ of a Hilbert space, an operator $T_xi$ is defined as the operator associated to a sesquilinear form determined by $xi$. This operator is in general different to the classical frame operator but possesses some remarkable properties. For instance, $T_xi$ is self-adjoint (in an specific space), unconditionally defined and, when $xi$ is a lower semi-frame, $T_xi$ gives a simple expression of a dual of $xi$. The operator $T_xi$ and lower semi-frames are studied in the context of sequences of integer translates.