Isomorphisms of Orlicz spaces

Abstract

Abstract In this paper, we provide some isomorphism preserving conditions for (weighted) Orlicz spaces, and as a main result, it is proved that if there exist a bicontinuous linear operator T : L w 1 Φ ⁢ ( G 1 ) → L w 2 Φ ⁢ ( G 2 ) {T\colon L^{\Phi}_{w_{1}}(G_{1})\rightarrow L^{\Phi}_{w_{2}}(G_{2})} and a mapping a ↦ ( ξ ⁢ ( a ) , h ⁢ ( a ) ) {a\mapsto(\xi(a),h(a))} from G 1 {G_{1}} to ℂ × G 2 {\mathbb{C}\times G_{2}} with T ⁢ λ a = ξ ⁢ ( a ) ⁢ λ h ⁢ ( a ) ⁢ T {T\lambda_{a}=\xi(a)\lambda_{h(a)}T} for all a ∈ G 1 {a\in G_{1}} , then G 1 {G_{1}} and G 2 {G_{2}} are isomorphic, where Φ is a Δ 2 {\Delta_{2}} -regular Young function, G 1 {G_{1}} and G 2 {G_{2}} are locally compact groups and w 1 {w_{1}} and w 2 {w_{2}} are weight functions. Also, for a class of Young functions Φ, we show that if C ⁢ V Φ ⁢ ( G 1 ) {CV_{\Phi}(G_{1})} and CV Φ ⁢ ( G 2 ) {\mathrm{CV}_{\Phi}(G_{2})} are isometrically isomorphic, then G 1 {G_{1}} and G 2 {G_{2}} are isomorphic, CV Φ ⁢ ( G i ) {\mathrm{CV}_{\Phi}(G_{i})} is the space of all convolution operators on the Orlicz space L Φ ⁢ ( G i ) {L^{\Phi}(G_{i})} for i = 1 , 2 {i=1,2} .