Contractive projections in variable Lebesgue spaces

In this article we describe the structure of positive contractive projections in variable Lebesgue spaces Lp(·) with σ-finite measure and essentially bounded exponent function p(·). It is shown that every positive contractive projection P:Lp(·)→Lp(·) admits a matrix representation, and the restriction of P on the band, generated by a weak order unite of its image, is weighted conditional expectation operator. Simultaneously we get a description of the image R(P) of the positive contractive projection P. Note that if measure is finite and exponent function p(·) is constant, then the existence of a weak order unit in R(P) is obvious. In our case, the existence of the weak order unit in R(P) is not evident and we build it in a constructive manner. The weak order unit in the image of positive contractive projection plays a key role in its representation.