The \(K\)-dual Bessel generators for unitary groups of Hilbert spaces
Let \(K\) be a bounded operator on a separable Hilbert space \({\mathcal H}.\) A sequence \(\left(f_j\right)\) in \({\mathcal H}\) is said to be a \(K\)-frame if there exist constants \(0 < A \leq B < \infty\) such that \[ A\|K^\ast f\|^2 \leq \sum_j|\langle f, f_j\rangle|^2 \leq B\|f\|^2\ \ \forall f\in {\mathcal H}. \] The notion of \(K\)-frame was proposed by \textit{L. Găvruţa} in [Appl. Comput. Harmon. Anal. 32, 139--144 (2012; Zbl 1230.42038)]. Let \({\mathcal U}\) be a group of unitary operators on \({\mathcal H}.\) A \(K\)-frame generator for \({\mathcal U}\) is a vector \(\eta\in {\mathcal H}\) such that \({\mathcal U}\eta = \{U\eta:\ U\in {\mathcal U}\}\) is a \(K\)-frame of \({\mathcal H}.\) A vector \(\xi\in {\mathcal H}\) is a \(K\)-dual Bessel generator of \(\eta\) if \({\mathcal U}\xi\) is a Bessel sequence and \[ Kf = \sum_{U\in {\mathcal U}}\langle f, U\xi\rangle U\eta\ \ \forall f\in {\mathcal H}. \] The present paper contains two main results. The first one gives a complete description of all \(K\)-dual Bessel generators of \(\eta\) for \({\mathcal U}\) under two cases. The second one describes the minimal \(\ell^2({\mathcal U})\)-norm representation \(Kf = \sum_{U\in {\mathcal U}}c_U U\eta\) for \(f\in {\mathcal H}\) with \(c\in \ell^2({\mathcal U}).\)
scientific article; zbMATH DE number 7138394
scientific article