Exposed and smooth points of some classes of operation in L(lp)

AbstractLet X be a Banach space and B1(X) the unit ball of X. A point x of B1(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 B1(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 B1(L(lp)) and B1(K(lp)), where L(lp) is the space of bounded linear operators on lp, 1 ⩽ p < ∞, and K(lp) is the class of compact operators in L(lp). Indeed, we characterize smooth points and exposed points of B1L(lp)) and B1(K(lp)) for 1 ⩽ p < ∞. Further, if 1 < p ≠ 2 < ∞ we show that not every extreme point of B1(L(lp)) is exposed. We also characterize the exposed points of B1(L(l1, lp)), 1 ⩽ p < ∞