We prove that a tensor norm a (defined on tensor products of Hilbert spaces) is the Hilbert-Sehmidt norm if and only if l_2 (direct X) … (direct X) l_2. endowed with the norm a, has an unconditional basis. This extends a classical result of Kwapien and Petczyriski. The symmetric version of that statement follows, and this extends a recent result of Defant, Diaz, Garcia and Maestre.
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