gives rise to a well-defined propagator, which is a semigroup of linear operators, and the theory of semigroups of linear operators on Banach spaces has developed quite rapidly since the discovery of the generation theorem byHille and Yosida in 1948. By now, it is a rich theory with substantial applications to many fields cf., e.g., 1–6 . In this paper, we pay attention to some basic problems o...

Let G be a semigroup of commuting linear operators on a linear space S with the group operation of composition. The solvability of the system of equations /,/ = ,■, i = I, ... , r, where /, e G and j e S , was considered by Dahmen and Micchelli in their studies of the dimension of the kernel space of certain linear operators. The compatibility conditions Ijfr = Irfj , i ^ j , are necessar...

The paper is devoted to the problem of existence of propagators for an abstract linear non-autonomous evolution Cauchy problem of hyperbolic type in separable Banach spaces. The problem is solved using the so-called evolution semigroup approach which reduces the existence problem for propagators to a perturbation problem of semigroup generators. The results are specified to abstract linear non-...

For convenience, we can have a look at the list of materials contained in this lecture first. Note here we always consider the time-homogenous Markov processes. • Transition kernels μt, t ≥ 0 ←→ transition operators semigroup Tt, t ≥ 0. • Feller semigroup Tt, t ≥ 0 • Feller processes • existence • every Feller process has a cadlag version • regularization theorem for submartingale • strong Mark...

We consider properties of second-order operatorsH = − ∑d i,j=1 ∂i cij ∂j on R with bounded real symmetric measurable coefficients. We assume that C = (cij) ≥ 0 almost everywhere, but allow for the possibility that C is singular. We associate with H a canonical self-adjoint viscosity operator H0 and examine properties of the viscosity semigroup S generated by H0. The semigroup extends to a posit...

The diamagnetic inequality for the magnetic Schrödinger semigroup is extended to the difference of the semigroups of magnetic Schrödinger operators with Neumann and Dirichlet boundary conditions on arbitrary open domains and rather general magnetic vector potentials A and potentials V . In particular, this bound renders moot all the technical issues in the recent proofs of the independence of t...

Abstract Molecular Hamiltonians are constituted by momentum operators (e.g. kinetic energy and interaction with electro-magnetic radiation), local potential operators (e.g. electron-electron coulomb, electron-nucleus and electron-external field interactions) and spin operators. In general the effect of transformations that generate states from a reference state is to destroy this form. Here it ...