Banach Space Problem

Let E be an arbitrary Banach space and let T:E*-->l_2 be a linear continuous operator. Is it true that T must be the so-limit (i.e., pointwise limit) of a net (S_d)* of adjoint opetators, with S_d:l_2-->E and ||S_d||<=||T|| ?

I think not, say E=c_0(I) and T is a surjection, where I has a "big" cardinality.

Any help will be highly appreciated.
Ady.

P.S. Of course, here l_2 means the usual Hilbert sequence space, and E* denotes the dual of E.
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