Complex extreme points and complex rotundity in Orlicz spaces equipped with the 𝑠-norm

Abstract

Let Φ be an Orlicz function and 𝐿 Φ(𝑋, Σ, 𝜇) be the corresponding Orlicz space on a non-atomic, 𝜎-finite, complete measure space (𝑋, Σ, 𝜇). It is known that extreme points which are connected with rotundity of the whole spaces are the most essential and important geometric notion in the geometric theory of Banach spaces. On the other hand, geometric theory of complex Banach spaces has significant applications that differ from the geometric theory of real Banach spaces. In this paper, we first describe the complex extreme points of unit ball of Orlicz spaces equipped with the 𝑠-norm where 𝑠 is a strictly increasing outer function. We also give criteria for complex rotundity. Our study generalizes and unifies the results that have been obtained for the Orlicz norm and the 𝑝-Amemiya norm (1 < 𝑝 < ∞) separately.

Keywords

• Orlicz space
• complex extreme points
• complex strictly rotund
• s-norm

References

1. Arıs, B., Öztop, S., 2023, Wiener amalgam spaces with respect to Orlicz spaces on the affine group. J. Pseudo-Differ. Oper. Appl., 14(23) google scholar
2. Başar E., Öztop, S., Uysal, B.H., Yaşar, Ş. 2023, Extreme points in Orlicz spaces equipped with 𝑠-norms and its closedness, Math. Nachr. 00, 1– 21. google scholar
3. Chen, L., Cui, Y., 2010, Complex extreme points and complex rotundity in Orlicz function spaces equipped with the p-Amemiya norm, Nonlinear Anal., 73(5), 1389-1393. google scholar
4. Chen, S., 1996, Geometry of Orlicz space. Dissertationes Math., Harbin University, China. google scholar
5. Cui, Y., Duan, L., Hudzik, H. and Wisła, M., 2008, Basic theory of 𝑝-Amemiya norm in Orlicz spaces (1 ≤ 𝑝 ≤ ∞): Extreme points and rotundity in Orlicz spaces endowed with these norms, Nonlinear Anal., 69(5), 1796-1816. google scholar
6. Cui, Y., Zhan, Y., 2019, Strongly extreme points and middle point locally uniformly convex in Orlicz spaces equipped with s-norm, Journal of Funct. Spac., 2019, 1-7. google scholar
7. Orlicz, W., 1932, Über eine gewisse klasse von Räumen vom typus B, Bull. Int. Acad. Polon. Sér. A, 207-220. google scholar
8. Rao, M. M. and Ren, Z. D., 1991, Theory of Orlicz Spaces, Marcel Dekker, New York. google scholar

9. Üster, R., 2021, Multipliers for the weighted Orlicz spaces of a locally compact abelian group, Results in Mathematics, 76 (4), Paper No. 183. google scholar
10. Thorp, E., Whitley, R., 1967, The strong maximum modulus theorem for analytic functions into a Banach space, Proc. Amer. Math. Soc., 18, 640-646. google scholar
11. Wisła, M., 2020, Orlicz spaces equipped with 𝑠-norms, J. Math. Anal. Appl., 483(2), 123659-123689. google scholar
12. Wu, C.X., Sun, H., 1987, On complex extreme points and complex strict convexities of Musielak–Orlicz spaces, J. Sys. Sci. Math. Scis., 7, 7-13. google scholar
13. Wu, C.X., Sun, H., 1988 On complex uniform convexity of Musielak–Orlicz spaces, J. Northeast. Math., 4, 389-396. google scholar