Contractive projections in variable Lebesgue spaces

In this article we describe the structure of positive contractive projections in variable Lebesgue spaces $L_{p(\cdot)}$ with $\sigma$-finite measure and essentially bounded exponent function $p(\cdot)$. It is shown that every positive contractive projection $P:L_{p(\cdot)}\rightarrow L_{p(\cdot)}$ 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 $\mathcal{R}(P)$ of the positive contractive projection $P$. Note that if measure is finite and exponent function $p(\cdot)$ is constant, then the existence of a weak order unit in $\mathcal{R}(P)$ is obvious. In our case, the existence of the weak order unit in $\mathcal{R}(P)$ 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.