Operators with the Lipschitz bounded approximation property (Q6174100)

The present paper is dedicated to the Lipschitz bounded approximation property for bounded linear operators. The authors show that whenever an operator can be approximated by a net of uniformly bounded finite rank Lipschitz mappings pointwise, then it can also be approximated by a net of uniformly bounded finite rank linear operators under the strong operator topology. As a consequence of it, they recover a theorem due to \textit{G.~Godefroy} and \textit{N.~J. Kalton} [Stud. Math. 159, No.~1, 121--141 (2003; Zbl 1059.46058)] which says that the Lipschitz bounded approximation property and the bounded approximation property are equivalent for every Banach space. Another application of their main result implies that a Banach space has an (unconditional) Lipschitz frame if and only if it has an (unconditional) Schauder frame.