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 $(\mathcal{H},⟨\cdot ,\cdot ⟩)$, the operator $T_{\xi }$ is defined as the operator associated to the sesquilinear form ${\Omega }_{\xi }(f,g)={\sum }_{n\in \mathbb{N}}⟨f,{\xi }_n⟩⟨{\xi }_n,g⟩,$ for $f,g\in \{h\in \mathcal{H}:{\sum }_{n\in \mathbb{N}}|⟨h,{\xi }_n⟩{|}^2<\infty \}$. 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.

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