Complex Convexity of Musielak-Orlicz Function Spaces Equipped with thep-Amemiya Norm

Lili Chen,Yanfeng Zhao,Yunan Cui

doi:10.1155/2014/190203

Lili Chen, Yanfeng Zhao + Show 1 more

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https://doi.org/10.1155/2014/190203

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Abstract

The complex convexity of Musielak-Orlicz function spaces equipped with thep-Amemiya norm is mainly discussed. It is obtained that, for any Musielak-Orlicz function space equipped with thep-Amemiya norm when1≤p<∞, complex strongly extreme points of the unit ball coincide with complex extreme points of the unit ball. Moreover, criteria for them in above spaces are given. Criteria for complex strict convexity and complex midpoint locally uniform convexity of above spaces are also deduced.

Highlights

Let (X, ‖ ⋅ ‖) be a complex Banach space over the complex field C, let i be the complex number satisfying i2 = −1, and let B(X) and S(X) be the closed unit ball and the unit sphere of X, respectively
In the early 1980s, a huge number of papers in the area of the geometry of Banach spaces were directed to the complex geometry of complex Banach spaces
It is well known that the complex geometric properties of complex Banach spaces have applications in various branches, among others in Harmonic Analysis Theory, Operator Theory, Banach Algebras, C∗Algebras, Differential Equation Theory, Quantum Mechanics Theory, and Hydrodynamics Theory

Summary

Introduction

The complex convexity of Musielak-Orlicz function spaces equipped with the p-Amemiya norm is mainly discussed. For any Musielak-Orlicz function space equipped with the p-Amemiya norm when 1 ≤ p < ∞, complex strongly extreme points of the unit ball coincide with complex extreme points of the unit ball. In [7], Thorp and Whitley first introduced the concepts of complex extreme point and complex strict convexity when they studied the conditions under which the Strong Maximum Modulus Theorem for analytic functions always holds in a complex Banach space.

Results

Conclusion