Power-norms based on Hilbert $C^{\ast }$-modules

Abstract

Suppose that $\mathcal{E}$ and $\mathcal{F}$ are Hilbert $C^{\ast }$-modules. We present a power-norm $\left({\Vert \cdot \Vert }_n^{\mathcal{E}}:n\in \mathbb{N}\right)$ based on $\mathcal{E}$ and obtain some of its fundamental properties. We introduce a new definition of the absolutely $(2,2)$-summing operators from $\mathcal{E}$ to $\mathcal{F}$, and denote the set of such operators by ${\tilde{\Pi }}_2(\mathcal{E},\mathcal{F})$ with the convention ${\tilde{\Pi }}_2(\mathcal{E})={\tilde{\Pi }}_2(\mathcal{E},\mathcal{E})$. It is known that the class of all Hilbert--Schmidt operators on a Hilbert space $\mathcal{H}$ is the same as the space ${\tilde{\Pi }}_2(\mathcal{H})$. We show that the class of Hilbert--Schmidt operators introduced by Frank and Larson coincides with the space ${\tilde{\Pi }}_2(\mathcal{E})$ for a countably generated Hilbert $C^{\ast }$-module $\mathcal{E}$ over a unital commutative $C^{\ast }$-algebra. These results motivate us to investigate the properties of the space ${\tilde{\Pi }}_2(\mathcal{E},\mathcal{F})$.

22 pages, Accepted by Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM