Affine semigroups on Banach spaces
On a Banach space \(X\) the one-parameter family \(\{S(t),\,t\geq 0\}\) of affine transformations with the properties \((S_ 1)\) \(S(0)=I\), \(S(t+S)=S(t)\cdot S(s)\), \(s,t\geq 0\); and \((S_ 2)\) \(\forall x\in X\), \(t\mapsto S(t)x\) is a continuous function from \([0,\infty)\) into \(X\) is considered. Such a family is called an affine semigroup. An affine semigroup \(S(t)\) decomposes into a linear part \(T(t)\) and a translation part \(z(t)\) by formulas \[ z(t)=S(t)0,\quad T(t)x=S(t)x-S(t)0. \] Here \(T(t)\) is a \(C_ 0\)-semigroup of linear operators and the function \(z(t): [0,\infty)\to X\) satisfies the equality \[ z(t+s)=T(t)z(s)+z(t),\quad t,s\geq 0. \] On a Banach space \(\hat X=X\times R\) the augmented semigroup \[ \hat T(t)[x,\tau]=[T(x)x+\tau z(t),\tau],\quad [x,\tau]\in X \] associated with \(S(t)\) is constructed. The relations between the infinitesimal generators \(A,U,\hat U\) of the semigroups \(S(t)\), \(T(t)\), \(\hat T(t)\), respectively, and the linear and translation parts of the semigroup \(S(t)\) are established. The necessary and sufficient conditions for the function \(z(\cdot):[0,\infty)\to X\) to be the translation part of an affine semigroup and for the operator \(A: X\to X\) to be the infinitesimal generator of such a semigroup are studied. Analogous results for the affine cosine functions are obtained. The criteria of compactness and analyticity for affine semigroups which are analogous to the linear case are formulated. The affine group is also considered.
scientific article; zbMATH DE number 4105664