Pair frames (Q471752)

For two sequences \(F=\{f_i\}\), \(G=\{g_i\}\) of elements of a Hilbert space \({\mathcal H}\) and a scalar sequence \(m=\{m_i\}\), the authors call the triplet \((m,G,F)\) an \textit{m-pair Bessel} for \({\mathcal H}\) if the operator \(S_{mFG}(f):=\sum_i m_i\langle f,g_i\rangle f_i\) is well--defined. In particular, if \(m=\{1\}\), they call \((G,F)\) a \textit{pair Bessel}. If, in addition, the operator \(S_{mFG}(f)\) is invertible they call it an \textit{m-pair frame}, and if \(m=\{1\}\), then \((G,F)\) is a \textit{pair frame}. In this paper the authors study some properties of certain kinds of \(m\)-pair Bessel and \(m\)-pair frames and show, in particular, that \(F\) is a frame (resp. Bessel sequence) for \({\mathcal H}\) if and only if \((F,F)\) is a pair frame (resp. pair Bessel) for \({\mathcal H}\). If \((m,G,F)\) is an \(m\)-pair Bessel, Balasz [\textit{P. Balazs}, J. Math. Anal. Appl. 325, No. 1, 571--585 (2007; Zbl 1105.42023)] called \(S_{mFG}\) a \textit{multiplier operator}. The invertibility of these operators was studied by \textit{D. T. Stoeva} and \textit{P. Balazs} [Appl. Comput. Harmon. Anal. 33, No. 2, 292--299 (2012; Zbl 1246.42017)].