Von Neumann-Schatten frames in separable Banach spaces
A sequence \(\{\Lambda_i\in \mathcal{B}(\mathcal{H}, \mathcal{K}_i) : i\in\mathbb {N}\}\) is a \(g\)-frame for \(H\) with respect to \(\{\mathcal{K}_i : i\in\mathbb{N}\}\) if there are two positive constants \(A\) and \(B\) such that \[ A\|f\|^2\leq \sum_{i\in I}\|\Lambda_i f\|^2\leq B\|f\|^2~ (f\in\mathcal{H}). \] A sequence \(\{g_i\}_{i=1}^{\infty}\subset \mathcal{X}^*\) is called a \(p\)-frame \((1<p<\infty)\) for \(\mathcal{X}\) if there exist two positive constants \(A\) and \(B\) so that \[ A\|f\|\leq \left(\sum_{i=1}^{\infty} |g_i(f)|^p\right)^{1/p}\leq B\|f\|~ (f\in \mathcal{X}). \] Let \(\mathcal{C}_p\) be the von Neumann-Schatten \(p\)-class. Let \((\mathcal{G}_i)_{i=1}^{\infty}\) be a sequence of bounded linear operators from \(\mathcal{X}\) to \(\mathcal{C}_p\subset \mathcal{B}(\mathcal{H})\). We say that \((\mathcal{G}_i)_{i=1}^{\infty}\) is a von Neumann-Schatten \(p\)-frame for \(\mathcal{X}\) with respect to \(\mathcal{H}\) if there exist two positive constants \(A\) and \(B\) such that \[ A\|f\|\leq \left(\sum_{i=1}^{\infty} \| \mathcal{G}_i(f)\|_p^{p}\right)^{1/p}\leq B\|f\|~ (f\in \mathcal{X}). \] It is shown that \(p\)-frames and \(g\)-frames are a class of von Neumann-Schatten \(p\)-frames.
scientific article; zbMATH DE number 6107826
scientific article