Norm derivatives on spaces of operators

Abstract

We say the norm is Frechet differentiable at x if the convergence to the limit in (1.1) is uniform for all y~X. Norm derivatives of the spaces: LP(dy), 1 ~ p N oo, and C(X) space of real continuous functions on a compact Hausdorff space X have been studied extensively, see for example page 171 in [5]. Also necessary and sufficient conditions for the existence of these derivatives in an arbitrary Banach space X have been established in terms of support functionals in the dual space X*, see [2]. In this paper we investigate the existence of norm derivatives in the spaces of compact operators % on a separable Hilbert space whose norm will be defined in the following section. We also give necessary and sufficient conditions for the existence of the Gateaux derivative of the norm in B(H): the space of bounded operators, on a Hilbert space H, with the uniform norm. Whenever the derivative exists we write a formula for it. Norm derivatives are important in approximation theory and in the geometry of Banach spaces. We will make use of the following basic theorem in Corollary 3.3. See [5], ex. 3.57.