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

Abstract

Given an arbitrary sequence of elements $\xi=\{\xi_n\}_{n\in \mathbb{N}}$ of a Hilbert space $(\mathcal{H},\langle\cdot,\cdot\rangle)$, the operator $T_\xi$ is defined as the operator associated to the sesquilinear form $ \Omega_\xi(f,g)=\sum_{n\in \mathbb{N}} \langle f,\xi_n\rangle\langle\xi_n,g\rangle, $ for $f,g\in \{h\in \mathcal{H}: \sum_{n\in \mathbb{N}}|\langle h,\xi_n\rangle|^2<\infty\}$. This operator is in general different from the classical frame operator but possesses some remarkable properties. For instance, $T_\xi$ is always self-adjoint in regards to a particular 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 of a function of $L^2(\mathbb{R})$. In particular, an explicit expression of $T_\xi$ is given in this context and a characterization of sequences of integer translates which are lower semi-frames is proved.