On nonnegative solvability of linear operator equations
Let \(E\) be a Banach lattice with order continuous norm and let \(T\) be a positive reducible operator, some power of which is compact. Extending earlier work of the authors in the finite-dimensional setting and in the Lebesgue spaces \(L^ p(\Omega,\Sigma,\mu)\), \(1\leq p< \infty\), they find necessary and sufficient conditions for the equation \(\lambda x= Tx+ y\), \(y>0\), \(\lambda>0\), to have a positive solution \(x\). In their analysis a Frobenius decomposition of \(T\) with respect to a decomposition of \(E\) into disjoint bands is used to induce a corresponding decomposition of the adjoint operator \(T^*\). This is used to analyse the structure of the adjoint algebraic eigenspaces pertaining to distinguished eigenvalues of \(T\) (an eigenvalue is called distinguished, if it is positive and has a positive eigenvector associated with it). Three of the four conditions found are: (a) for any \(f^*\) in the adjoint algebraic eigenspace to any distinguished eigenvalue \(\lambda_ i\) of \(T\), \(\lambda_ i\geq \lambda\), it follows that \(f^*(y)= 0\); (b) the norm and order limit of \(\sum^ m_{j=0} \lambda^{-j} T^ j y\) as \(m\to\infty\) exists in \(E\); and (c) \(\lim_{m\to \infty} \lambda^{-m} T^ m y=0\) in both the norm and order senses.
Banach lattice with order continuous norm
scientific article; zbMATH DE number 515557
scientific article