On the sublinear functional associated to a family of invariant means
Conditions are given under which explicit analytic expressions can be obtained for the fundamental sublinear functional \(\rho_ K(x)=\sup \{\mu (x):\) \(\mu\in K\}\) associated to a family of means \(K=[G]\cap J\), where \(x\in \ell^{\infty}\), G is a subset of means on \(\ell^{\infty}\) invariant with respect to a semigroup of operators G and J is a saturated set of means on \(\ell^{\infty}\). Under such conditions it is possible to assure that \(K\neq \emptyset.\) A relation is also obtained between \(\rho_ K\) and \(\rho_ J\) which allows to give \(\rho_ K\) in terms of \(\rho_ J\) when \(\rho_ J\) is known and different equivalent expressions for \(\rho_ K\) if the family J changes and K is fixed. From this several examples are given.
scientific article; zbMATH DE number 3964769
scientific article