Semigroups of difference operators in spectral analysis of linear differential operators (Q1357902)

Let \(U\) denote a family of evolution operators for the equation \(x(t)= A(t)x(t)\), \(t\in\mathbb{R}\), where \(A(t): D(A(t))\subset X\to X\) is a family of closed linear operators that generate a correct Cauchy problem; \(X\) is a complex Banach space. To the family \(U\) it is assigned a linear operator \[ L_U: D(L_U)\subset F\to F, \] where \(F\) may be one of the four Banach spaces defined in the paper. The domain \(D(L_U)\) is defined as follows. A function \(x\in F\) belongs to \(D(L_U)\) iff there exists a function \(f\in F\) such that for almost all \(s,t\in\mathbb{R}\), with \(s\leq t\) one has \[ x(t)= U(t,s) x(s)- \int^t_s U(t,\tau) f(\tau)d\tau. \] Due to these definitions for the operator \(L_U\) there holds \[ L_U= -{d\over dt}+ A(t): D(L_U)\subset F\to F \] is an abstract parabolic operator, also \(L_Ux= f\). Several interesting results on the operator \(L_U\) are embodied in the four theorems of the paper. In that, the semigroup of difference operators \((T_U(t)x)(s)= U(s,s- t) x(s- t)\), \(x\in F\), \(s\in\mathbb{R}\), \(t\geq 0\) is used.