A minimax theorem for irreducible compact operators in \(L^ p\)-spaces
type: Item
instance of: scholarly article
title: A minimax theorem for irreducible compact operators in \(L^ p\)-spaces
zbMATH Open document ID: 0576.47020
DOI: 10.1007/BF02761164
published in: Israel Journal of Mathematics
publication date: 1984-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 47B60
Mathematics Subject Classification ID: 47A10
Mathematics Subject Classification ID: 47B38
zbMATH DE Number: 3921316
zbMATH Keywords: irreducible compact operator
zbMATH Keywords: spectral radius
zbMATH Keywords: minimax property
zbMATH Keywords: irreducible finite rank operators
author: Helmut H. Schaefer
MaRDI profile type: Publication
cites work: Q4110164
cites work: Q5531984
cites work: Topologische Nilpotenz irreduzibler Operatoren
cites work: Q4049667
cites work: Unzerlegbare, nicht negative Matrizen
cites work: Compactness of integral operators in Banach function spaces
description: scientific article; zbMATH DE number 3921316