IMAGES OF OPERATORS IN REARRANGEMENT INVARIANT SPACES AND INTERPOLATION

For every closed set G ⊂ [0, 1] there exists an injective linear operator T = T_G bounded on L₁[0, 1] and L_∞[0, 1] such that T is normally solvable in an Orlicz space L_M[0, 1] if and only if G ∩ [α_M, β_M] = ∅ (α_M and β_M are the Boyd indices of the space L_M). Analogous result holds also for isomorphisms of spaces of functions defined on the semiaxis [0, ∞).