On a new geometric property for Banach spaces

Abstract

In this paper we study a geometric property for Banach spaces called condition (∗), introduced by de Reynaet al in [3]. A Banach space has this property if for any weakly null sequence {x_n} of unit vectors in X, if {x^∗_n} is any sequence of unit vectors in X^∗ that attain their norm at x_n's, then x^n∗ → 0. We show that a Banach space satisfies condition (∗ ) for all equivalent norms iff the space has the Schur property. We also study two related geometric conditions, one of which is useful in calculating the essential norm of an operator.