Bessel fusion multipliers
Let \(H\) be a separable Hilbert space. A sequence \((\psi_k)\) in \(H\) is called a Bessel sequence if there exists a positive constant \(B\) for which \[ \sum\limits_{k=1}^\infty |\langle \xi, \psi_k \rangle |^2\leq B ||\xi ||^2 \] for all \(\xi \in H\). Given two Bessel sequence \((\phi_k)\) and \((\psi_k)\) in \(H_1\) and \(H_2\) and a fixed sequence \(m=(m_k) \in\ell^\infty\), the following rescaling operator is defined \[ M_m(\psi_k)(\phi_k) (\xi)=\sum\limits_{k=1}^\infty m_k \langle \xi, \phi_k \rangle \psi_k. \] This operator is called the Bessel multiplier. The authors study properties of such multipliers when \(m\in \ell^p\). A more general concept of Bessel fusion multiplier is introduced and investigated for \(m\in \ell^p\).
scientific article; zbMATH DE number 5355958
scientific article