A complete characterization of smoothness in the space of bounded linear operators

Abstract

We completely characterize smoothness of bounded linear operators between infinite dimensional real normed linear spaces, probably for the very first time, by applying the concepts of Birkhoff-James orthogonality and semi-inner-products in normed linear spaces. In this context, the key aspect of our study is to consider norming sequences for a bounded linear operator, instead of norm attaining elements. We also obtain a complete characterization of smoothness of bounded linear operators between real normed linear spaces, when the corresponding norm attainment set non-empty. This illustrates the importance of the norm attainment set in the study of smoothness of bounded linear operators. Finally, we prove that Gâteaux differentiability and Fréchet differentiability are equivalent for compact operators in the space of bounded linear operators between a reflexive Kadets-Klee Banach space and a Fréchet differentiable normed linear space, endowed with the usual operator norm.2010 Mathematics Subject Classification. Primary 46B20, Secondary 47L05.

Numerical radius and a notion of smoothness in the space of bounded linear operators

“…The main objective of the present article is to study smoothness in the space of bounded linear operators on a Banach space, induced by the numerical radius. The study of smoothness in the space of bounded linear operators on a Banach space, with respect to the usual operator norm, is a classical area of research in geometry of Banach spaces [1,[6][7][8]12,[15][16][17][18]24,25]. The space of bounded linear operators on a Banach space, endowed with the numerical radius norm, need not be isometrically isomorphic to the space of bounded linear operators on the same Banach space, endowed with the usual operator norm, in general.…”

“…In particular, the norm attainment set of a linear operator T plays a pivotal role in determining the smoothness of T in L(X). A characterization of smoothness without any restriction on M T was provided in [24]. The above studies motivate us to explore the concept of numerical radius smoothness in (L(X)) w .…”

Birkhoff-James extensions of continuous functions on metric spaces

“…James [3,4] used it extensively in his study of geometric properties of normed linear spaces like smoothness and strict convexity. Recent studies [6,1,8,12,11] of this phenomenon in the context of operator theory have revealed a lot about the geometry of linear operators. Indeed, if T and U be a pair of bounded linear operators on a Banach space B over F(R or C), we say T ⊥ B U if ||T + λU || ≥ ||T || for all λ ∈ F, where ||.|| denotes the usual operator norm.…”

Characterization of k-smooth operators between Banach spaces

“…To do so we will use norm attainment set of an operator defined as : For T ∈ L(X, Y), the norm attainment set, denoted as M T , is the collection of all unit vectors at which T attains its norm, i.e., M T = {x ∈ S X : T x = T }. To look into the properties of norm attainment set and its role in the study of smoothness of operators one may go through [11,12,15,17].…”