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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.
Let $Sinmathcal{M}_d(mathbb{C})^+$ be a positive semidefinite $dtimes d$ complex matrix and let $mathbf a=(a_i)_{iinmathbb{I}_k}in mathbb{R}_{>0}^k$, indexed by $mathbb{I}_k={1,ldots,k}$, be a $k$-tuple of positive numbers. Let $mathbb T_{d}(mathbf a
We develop a natural generalization of vector-valued frame theory, we term operator-valued frame theory, using operator-algebraic methods. This extends work of the second author and D. Han which can be viewed as the multiplicity one case and extends
For a given finitely generated shift invariant (FSI) subspace $cWsubset L^2(R^k)$ we obtain a simple criterion for the existence of shift generated (SG) Bessel sequences $E(cF)$ induced by finite sequences of vectors $cFin cW^n$ that have a prescribe
In this paper we consider two problems in frame theory. On the one hand, given a set of vectors $mathcal F$ we describe the spectral and geometrical structure of optimal completions of $mathcal F$ by a finite family of vectors with prescribed norms,
It is known in Hilbert space frame theory that a Bessel sequence can be expanded to a frame. Contrary to Hilbert space situation, using a result of Casazza and Christensen, we show that there are Banach spaces and approximate Bessel sequences which c