On the spectral radius of positive operators
We present several conditions which guarantee that a positive operator on a Banach lattice has a strictly positive spectral radius. The conditions are given in terms of commutativity properties. Sample results: \(\bullet\) Every positive operator that commutes with an ideal irreducible compact positive operator has a positive spectral radius. \(\bullet\) Every positive kernel operator that commutes with an order continuous band irreducible positive operator has a positive spectral radius. Some of these results can be proven by formulating a lattice hyperinvariant subspace theorem à la Lomonosov. The obtained results are either new or far reaching extensions of well known theorems. In the process of establishing our results, we also introduce several new classes of positive operators which satisfy a variety of irreducibility properties. These classes of operators are of independent interest on their own and they are classified and illustrated with many examples.
order continuous band irreducible positive operator
scientific article; zbMATH DE number 126415
scientific article