Suppose that E and F are Hilbert C∗-modules. We present a power-norm (∥⋅∥En:n∈N) based on E and obtain some of its fundamental properties. We introduce a new definition of the absolutely (2, 2)-summing operators from E to F, and denote the set of such operators by Π~2(E,F) with the convention Π~2(E)=Π~2(E,E). It is known that the class of all Hilbert–Schmidt operators on a Hilbert space H is the same as the space Π~2(H). We show that the class of Hilbert–Schmidt operators introduced by Frank and Larson coincides with the space Π~2(E) for a countably generated Hilbert C∗-module E over a unital commutative C∗-algebra. These results motivate us to investigate the properties of the space Π~2(E,F).

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