A bounded approximation of weakly compact operators

We study bounded approximation properties of operators. It is shown that for every weakly compact operator TT on a Banach space XX, if TT is uniformly approximated on each compact subset of XX by finite rank operators, then T2T2 can be approximated in the same topology by finite rank operators with the operator norm less than or equal to ‖T‖2‖T‖2.