Compactness and essential norms of composition operators on Orlicz-Lorentz spaces
Let \((\Omega,\Sigma,\mu)\) be a \(\sigma\)-finite and complete measure space. In the first theorem (Theorem 2.1) in the paper under review, it is proved that the composition operator \(C_T\) is compact on the Orlicz-Lorentz space \(\Lambda^{\varphi, w}(\mu)\) if and only if, for each \(\varepsilon >0\), the set \[ N=N(h,\varepsilon)=\{x\in\Omega: h(x)>\varepsilon\},\;\text{where}\;h(x)=d\mu\circ T^{-1}(x)/\,d\mu(x), \] consists of only finitely many atoms. By assuming, in addition, that \(\Omega\) consists of countably many atoms, that \(\mu(\Omega)=\infty\), and that either the generating Orlicz function \(\varphi\) or its Young conjugate \(\varphi^*\) satisfies condition \(\Delta_2\) for all \(u\geq 0\), the following estimates for the essential norm of a composition operator \(C_T\) on \(\Lambda^{\varphi, w}(\mu)\) in terms of the value \[ \varepsilon_0=\inf\{\varepsilon >0 : N(h,\varepsilon) \text{ consists of only finitely many atoms}\} \] are obtained: (i) \(\|C_T\|_e \geq \varepsilon_0\) if \(0<\varepsilon_0\leq 1\). (ii) \(\|C_T\|_e \leq \varepsilon_0\) if \(\varepsilon_0 > 1\). (Note that by Theorem 2.1, \(\|C_T\|_e =0\) if and only if \(\varepsilon_0=0\).)
scientific article; zbMATH DE number 6094522
scientific article