Abstract
It is a well known fact that operators on a separable Hilbert space H giving norm-summability on an orthonormal basis have to be nuclear (Holub 1972) and operators giving summability on an orthonormal basis must be Hilbert-Schmidt. In former papers the author characterizes all the sequences of H that in this respect behave as orthonormal basis, and in the present paper those results are in some way, generalized to a separable Banach space.
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