Unitary Semigroup Research Articles

Chen-Möbius inversion theorem and a structural representation of crystallographic direction families

An alternative formula for the inverse calculation of the pairwise interatomic potential from the cohesive energy as a function of the lattice separation is presented. The formula is based on the structural representation of crystallographic direction families for a crystal lattice. The key mathematical treatment comes from the up-to-date Chen-Möbius inversion theorem. Both the case of a pure crystal lattice and the case of a binary superstructure are studied. The solution for the latter is based on a generalized Möbius inversion formula on a unitary semigroup. To make the technique of inverse calculation for pair potentials realistic and practicable for atomic simulations, the inverse calculation of the pair potential function with a volume-dependent correction term is discussed.

Stabilizability of two classes of contraction semigroups

This paper studies the strong stabilizability of two classes of Hilbert space contraction semigroups: (i) strict contraction semigroups, which include those with strictly dissipative generators; and (ii) isometric or unitary semigroups. The former class is already weakly stable, while the latter is not strongly stable over the whole space. Our tool is the functional model of Hilbert space contractions; hence, strong stability of the semigroup is studied via stability of its cogenerator. It is shown that a strict contraction semigroup is, in general, not strongly stabilized by the feedback −B*, while an isometric or a unitary semigroup is strongly stabilized by the same feedback, providedB is not compact.

On the reduction of a contraction semigroup to a completely non unitary semigroup

According to a Theorem of B. Sz.-Nagy and C. Foiaş, every strongly continuous semigroup of contraction operators on a Hilbert space, can be decomposed into a completely non unitary part and a unitary part. In this note we wish to show that by appropriately perturbing its generator, a contraction semigroup can be reduced to a completely non unitary one. In control theory, such a perturbation is related to the so called state feedback, and the reduction presented here has application in the problem of stabilizing linear control systems on a Hilbert space. This will be briefly discussed.

L1-Sequential Convergence on Stone Spaces and Stone's Theorem on Unitary Semigroups

The following theorem is a well-known tool in the study of measurable functions:Theorem. Let (M; μ) be a finite measure space and let (xn) be a sequence of functions in L1(M; μ) so that xn→0 in the norm of L1(M; μ). Then there is a sub-sequence so that xnk →0 pointwise almost everywhere on M.

A note on middle unitary semigroups

A note on middle unitary semigroups