Absolute Schur Algebras and Unbounded Matrices
Pachara Chaisuriya, Sing-Cheong Ong · SIAM Journal on Matrix Analysis and Applications · 1999
Let p, q, r be real numbers such that $p, q, r \ge 1$, and let $\B$ be a Banach algebra. Let $\Bl$ denote the set of all matrices which define bounded linear transformations from $\ell^p$ into $\ell^q$. The set $$ \SB = \Big\{A = \left[a_{jk}\right]: a_{jk} \in \B \ \mbox{ and} \ A^{[r]} = \left[\N{a_{jk}}^r\right] \in \mathcal{B}(\ell^p, \ell^q)\Big\} $$ of infinite matrices over $\B,$ is shown to be a Banach algebra under the Schur product operation, and the norm $\rn{A} = \|A^{[r]}\|^{1/r}$. For $r \ge 2$ and $\B = \mathbb{C}$, the complex field, $\mathcal{S}^p = \mathcal{S}^p(\mathbb{C})$ contains the set $\Bl$. For $r= 2$, $\mathcal{S}^2$ contains the bounded matrices $\mathcal{B}(\ell^p, \ell^q)$ as an ideal.