A minimax theorem for irreducible compact operators in \(L^ p\)-spaces
Let (X,\(\Sigma\),\(\mu)\) be a \(\sigma\)-finite measure space, T a compact irreducible (positive linear) operator on \(L^ p(\mu)\) \((1\leq p<+\infty)\). It is shown that the spectral radius r of T is characterized by the minimax property \[ \sup_{f\in Q}\inf_{E\in \Sigma_ 0}\int_{E}Tfd\mu /\int_{E}fd\mu =r=\inf_{fEQ}\sup_{E\in \Sigma_ 0}\int_{E}Tfd\mu /\int_{E}fd\mu. \] Where \(\Sigma_ 0\) denotes the ring of sets of finite measure and where Q denotes the set of all, almost everywhere positive functions in \(L^{\rho}\). Moreover, if \(r>0\) then equality on either side is assumed iff f is the (essentially unique) positive eigenfunction of T. Various refinements are given in terms of corresponding relations for irreducible finite rank operators approximating T.
scientific article; zbMATH DE number 3921316