Journal of Inequalities and Applications 
Volume 2 (1998), Issue 1, Pages 89-97
doi:10.1155/S1025583498000058

Von Neumann–Jordan constant for Lebesgue–Bochner spaces

Mikio Kato1 and Yasuji Takahashi2

 1Department of Mathematics, Kyushu Institute of Technology, Tobata, Kitakyushu 804, Japan
2Department of System Engineering, Okayama Prefectural University, Soja 719-11, Japan
Received 10 February 1997; Revised 23 May 1997

Abstract

The von Neumann–Jordan (NJ-) constant for Lebesgue–Bochner spaces Lp(X) is determined under some conditions on a Banach space X. In particular the NJ-constant for Lr(cp) as well as (cp) (the space of p-Schatten class operators) is determined. For a general Banach space X we estimate the NJ-constant of Lp(X), which may be regarded as a sharpened result of a previous one concerning the uniform non-squareness for Lp(X). Similar estimates are given for Banach sequence spaces lp(Xi) (lp-sum of Banach spaces Xi), which gives a condition by NJ-constants of Xi’s under which lp(Xi) is uniformly non-square. A bi-product concerning ‘Clarkson’s inequality’ for Lp(X) and lp(Xi) is also given.