Let (Ω,A,m) be a finite
measure space and Lp(1 < p < ∞) the Lebesgue space of all complex valued
measurable functions whose absolute p-th powers are integrable. Given a closed linear
subspace of Lp, the operator which assigns to f the function in the subspace with
minimum distance from it is continuous, idempotent, but not linear in general
except the case p = 2 when the operator is just an orthogonal projection. A
problem is to determine when such an operator Q is linear. It is linear if
and only if P = I − Q is a contractive projection, i.e., a linear idempotent
operator with ∥P∥≦ 1, so that the problem takes an equivalent form to
give complete description of contractive projections in Lp. In this paper the
problem will be settled in the latter form, not only for 1 < p < ∞ but also for
0 < p ≦ 1.
