Smooth perturbations in ordered Banach spaces and similarity for the linear transport operators
A new method for establishing a similarity for a pair of linear operators in an ordered Banach space is discussed. The author considers a pair of (in general unbounded) linear operators \(B_ 1\) and \(B_ 2=B_ 1+A\) with A a bounded operator, and assumes that both \(-B_ 1\) and \(-B_ 2\) generate bounded \(C_ 0\)-groups in the Banach space. He assumes, in addition, that \(e^{-tB_ 1}\) and -A are positivity preserving, and that \[ \int^{\infty}_{-\infty}\| Ae^{-tB_ 1}u\| dt<\infty \] for all u in the Banach space. He shows that \(B_ 2\) is similar to \(B_ 1\) i.e., \[ B_ 2=W_+(B_ 2,B_ 1)B_ 1W_+(B_ 2,B_ 1)^{-1} \] where \(W_+(B_ 2,B_ 1)=s-\lim_{t\to \infty}e^{tB_ 2}e^{-tB_ 1}\). The main idea depends on an analysis of positivity preserving \(C_ 0\)-groups. The method is applied to establish the similarity for the collisionless transport operator and the linear transport operator in multiple scattering problem in \(L^ 1\)- space.
similarity for a pair of linear operators in an ordered Banach space
multiple scattering problem in \(L^ 1\)-space
scientific article; zbMATH DE number 3969429