Invertible transformations acting on Orlicz spaces (Q1127566)

A Borel measurable function \(\tau: [0,1]\to [0,1]\) is called a transformation (with respect to the Lebesgue measure \(m\)) if \(m\circ\tau^{-1}\ll m\). Let \(G\) be the group of all invertible transformations \(\tau\). Let \(\Phi: \mathbb{R}\to \mathbb{R}_+\) be any Orlicz function satisfying the \(\Delta'\)-condition, i.e. \(\Phi(st)\leq c\Phi(s)\Phi(t)\) for certain \(c\) and all \(s,t\in\mathbb{R}\), and let \(L^\Phi(m)\) be the associated Orlicz space. With these notations the main results of the paper are the following: a) There is an injective embedding \(T^\Phi\) from \(G\) into the space \({\mathcal L}(L^\Phi(m))\) of all bounded linear operators on \(L^\Phi(m)\) given by \[ T^\Phi_\tau(f)= f\circ \tau^{-1}\omega_{\tau,\Phi}\quad\text{with}\quad \omega_{\tau,\Phi}= \Phi^{-1}\circ {d(m\circ \tau^{-1})\over dm}. \] b) The (metrizable) topologies \(\Theta_\Phi\) on \(G\) induced by \(T^\Phi\) and by the strong operator topology on \({\mathcal L}(L^\Phi(m))\) coincide for all \(\Phi\) and therefore, they coincide with \(\Theta_1\) associated to \(\Phi_1(s)=|s|\). c) Under an additional condition on \(\Phi\) it can be shown that \(\lim T^\Phi(G^3)\) is dense in \({\mathcal L}(L^\Phi(m))\). d) It is shown, that in contrast to the case \(\Phi_p(s)=|s|^p\) for general \(\Phi\) the map \(T^\Phi\) does not preserve the group action and does not map \(G\) into the group of isometries in \({\mathcal L}(L^\Phi(m))\).