A general Trotter--Kato formula for a class of evolution operators
From the authors' abstract: In this article, we prove new results concerning the existence and various properties of an evolution system \(U_{A+B}(t,s)_{0\leq s\leq t\leq T}\) generated by the sum \(-(A(t)+B(t))\) of two linear, time-dependent and generally unbounded operators defined on time-dependent domains in a complex and separable Banach space \(\mathcal B\). In particular, writing \(\mathcal L(\mathcal B)\) for the algebra of all linear bounded operators on \(\mathcal B\), we can express \(U_{A+B}(t,s)_{0\leq s\leq t\leq T}\) as the strong limit in \(\mathcal L(\mathcal B)\) of a product of the holomorphic contraction semigroups generated by \(-A(t)\) and \(-B(t)\), respectively, thereby proving a product formula of the Trotter--Kato type under very general conditions. We obtain a special case of our formula when \(B(t)=0\). We then illustrate our results by considering various examples of nonautonomous parabolic initial-boundary value problems, including one related to the theory of time-dependent singular perturbations of self-adjoint operators. We finally mention what we think remains an open problem for the corresponding equations of Schrödinger type in quantum mechanics.
scientific article; zbMATH DE number 5602676
scientific article