An interpolation result for the spectral radius (Q1110796)

Let be E a Banach lattice with order continuous norm and T a positive linear operator on E. Let EX(T) denote the family of all Banach lattices F, which contain a norm dense ideal \(F_ 0\) isomorphic to an order dense T-invariant ideal of E and satisfying \(\| T| F_ 0\|_ F<\infty\). In the main theorem of the paper it is proved that if \(F\in EX(T)\) and \[ \text{spectral radius}(T:F\to F)\leq \lambda \leq \text{spectral radius }(T:E\to E) \] then there exists a canonical \(H(\lambda)\in EX(T)\) with \[ \text{spectral radius}(T:H(\lambda)\to H(\lambda))=\lambda. \]