A minimax theorem for irreducible compact operators inL p-spaces

Let (X, Σ, μ) be a σ-finite measure space,T a compact irreducible (positive, linear) operator onL ^p (μ) (1≦p<+∞). It is shown that the spectral radiusr ofT is characterized by the minimax property {fx196-1} where ∑₀ denotes the ring of sets of finite measure and whereQ denotes the set of all, almost everywhere positive functions inL ^p. Moreover, ifr>0 then equality on either side is assumed ifff is the (essentially unique) positive eigenfunction ofT. Various refinements are given in terms of corresponding relations for irreducible finite rank operators approximatingT. Dedicated to H. G. Tillmann on his 60th birthday