Abstract
Let X be a Banach space and B 1( X) the unit ball of X. A point x of B 1( X) is called an exposed point if there exists f ϵ X ∗ such that ∥ f∥ = f( x) = 1 and f( y) < 1 for all y ≠ x in B 1( X). It is called smooth, if there exists a unique f ϵ X ∗ such that ∥ f∥ = f( x) = 1. The object of this paper is to study the exposed and smooth points of B 1( L( l p )) and B 1( K( l p )), where L( l p ) is the space of bounded linear operators on l p , 1 ⩽ p < ∞, and K( l p ) is the class of compact operators in L( l p ). Indeed, we characterize smooth points and exposed points of B 1 L( l p )) and B 1( K( l p )) for 1 ⩽ p < ∞. Further, if 1 < p ≠ 2 < ∞ we show that not every extreme point of B 1( L( l p )) is exposed. We also characterize the exposed points of B 1( L( l 1, l p )), 1 ⩽ p < ∞.
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