Operators with the Lipschitz bounded approximation property

We show that if a bounded linear operator can be approximated by a net (or sequence) of uniformly bounded finite rank Lipschitz mappings pointwisely, then it can be approximated by a net (or sequence) of uniformly bounded finite rank linear operators under the strong operator topology. As an application, we deduce that a Banach space has an (unconditional) Lipschitz frame if and only if it has an (unconditional) Schauder frame. Another immediate consequence of our result recovers the famous Godefroy-Kalton theorem (Godefroy and Kalton (2003)) which says that the Lipschitz bounded approximation property and the bounded approximation property are equivalent for every Banach space.