On the non-uniqueness of minimal projection in [formula omitted] spaces

Abstract

The main objective of this note is to exhibit a simple example of subspaces U ⊂ L p ( μ ) ( p ≠ 2 ) that admit two different projections with minimal norm. While for p = 1 , ∞ , such subspaces are well-known [W. Odyniec, G. Lewicki, Minimal Projections in Banach Spaces, in: Lecture Notes in Mathematics, vol. 1449, Springer-Verlag, Berlin, 1990. Problems of existence and uniqueness and their application], for 1 < p < ∞ their existence was open.