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

Suppose that $\mathcal{E}$ and $\mathcal{F}$ are Hilbert $C^{\ast }$-modules. We present a power-norm $({\parallel \cdot \parallel }_n^{\mathcal{E}}:n\in \mathbb{N})$ 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 ${\tilde{\mathrm{\Pi }}}_2(\mathcal{E},\mathcal{F})$ with the convention ${\tilde{\mathrm{\Pi }}}_2(\mathcal{E})={\tilde{\mathrm{\Pi }}}_2(\mathcal{E},\mathcal{E})$. It is known that the class of all Hilbert–Schmidt operators on a Hilbert space $\mathcal{H}$ is the same as the space ${\tilde{\mathrm{\Pi }}}_2(\mathcal{H})$. We show that the class of Hilbert–Schmidt operators introduced by Frank and Larson coincides with the space ${\tilde{\mathrm{\Pi }}}_2(\mathcal{E})$ 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 ${\tilde{\mathrm{\Pi }}}_2(\mathcal{E},\mathcal{F})$.