Some spectral properties of recurrent semigroups (Q1924904)

We say that a semigroup \(e^{-tA}\) on \(L^1(\Omega, \mu)\) is recurrent if for all \(f\in L^1(\Omega, \mu)_+\), \(f\not \equiv 0\) \[ Gf= \int^\infty_0 e^{-tA} f dt= \infty \quad (\mu-\text{a,e.}). \] We say that a semigroup \(e^{-tA}\) is irreducible if \(e^{-tA} f>0\) \((\mu-\) a.e.) for all \(f\in L^2 (\Omega, \mu)_+\), \(f\not \equiv 0\), \(t>0\). Definition. We say that \(e^{-tA}\) is \(L^1\)-stable if \(|e^{-tA}f|_1\to 0\) as \(t\to\infty\) for all \(f\in L^1(\Omega, \mu)\). The author proves the following result: Assume that \(e^{-tA}\) is irreducible. The following assertions are equivalent: i) \(e^{-tA}\) is recurrent; ii) \(e^{-t(A+V)}\) is \(L^1\)-stable for any \(V\geq 0\) \(V\not \equiv 0\) and \(e^{-tA}\) is not \(L^1\)-stable.