Generalized frame operator, lower semiframes, and sequences of translates

Abstract

Given an arbitrary sequence of elements $\xi =\lbrace \xi _n\rbrace _{n\in \mathbb {N}}$ 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 $f,g\in \lbrace h\in \mathcal {H}: \sum _{n\in \mathbb {N}}|\langle h , \xi _n\rangle |<^>2<\infty \rbrace$. 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 $L^2(\mathbb {R})$. In particular, an explicit expression of $T_\xi$ is given in this context, and a characterization of sequences of integer translates, which are lower semiframes, is proved.