Multismoothness in Banach Spaces

Abstract

In this paper, motivated by the results published by R. Khalil and A. Saleh in 2005, we study the notion ofk-smooth points and the notion ofk-smoothness, which are dual to the notion ofk-rotundity. Generalizing these notions and combining smoothness with the recently introduced notion of unitary, we study classes of Banach spaces for which the vector space, spanned by the state space corresponding to a unit vector, is a closed set.

k-Smooth Points in Some Banach Spaces

“…In [9] the authors characterized multismoothness in the completed injective tensor product ⊗ when is an 1 -predual space and is a smooth Banach space. We generalize their result to any Banach space .…”

“…Note that is a weak * -compact convex set and hence it is easy to see that ∈ -smooth ( ) if and only if dim(sp ext ) = . Multismoothness in Banach spaces was extensively studied by Lin and Rao in [9]. In paricular, they showed that, in a Banach space of finite dimension , any -smooth point is unitary and hence a strongly extreme point.…”

Characterization of k-smooth operators between Banach spaces

“…Following [8], we say that an element x ∈ S X is k−smooth or the order of smoothness of x is k, if J(x) contains exactly k linearly independent vectors, i.e., if k = dim span J(x). Similarly, an operator T ∈ L(X, Y) is said to be k−smooth operator if k = dim span J(T ), i.e., if there exist exactly k linearly independent functionals in S L(X,Y) * supporting the operator T. In [4,5,8,10,18], the authors have extensively studied k−smoothness in Banach spaces and in operator spaces. Though the characterization of k−smooth operators defined on Hilbert spaces [18] and between some particular Banach spaces are known, the complete characterization between arbitrary Banach spaces is still open.…”

$k$-smoothness on polyhedral Banach spaces

“…The study of k-smoothness plays an important role in identifying the structure of the unit ball of a Banach space. The papers [1,2,3,4] contain the study of k-smooth points of many Banach spaces. Several papers, including [1,3,4,5,6,7,9] study k-smoothness of operators on different spaces.…”

“…The papers [1,2,3,4] contain the study of k-smooth points of many Banach spaces. Several papers, including [1,3,4,5,6,7,9] study k-smoothness of operators on different spaces. In [7], the present authors have obtained a relation between k-smoothness and extreme points of the unit ball of a polyhedral Banach space.…”