Spectral properties of abstract parabolic operators
To every strongly continuous family \({\mathcal U}=\{{\mathcal U}(s,t)\colon 0\leq s \leq t<\infty\}\) of backward evolutionary operators in a complex Banach space \(X\) one can associate a linear operator \({\mathcal L}_{\mathcal U}\), taking a \(p\)-summable function \(x\colon {\mathbb R}_+\to X\) to the solution, \(f\), of the system of equations \[ x(s)={\mathcal U}(s,t)x(t) -\int_s^t {\mathcal U}(s,\tau)f(\tau) d\tau,\quad s\leq t \] (provided such a solution exists). This operator is closely related to the uniformly well-posed Cauchy problem on \({\mathbb R}_+\). Another object associated to the family \(\mathcal U\) is the semigroup \(T_{\mathcal U}(t)\), \(t\geq 0\) of linear difference operators in \(\text{End }(L_p({\mathbb R}_+,X))\), where \((T_{\mathcal U}(t)x)(s)={\mathcal U}(s,s+t)s(s+t)\). The paper under review relates the spectral properties of the operator \({\mathcal L}_{\mathcal U}\) to the properties of the semigroup \(T_{\mathcal U}(t)\), using the fact that \({\mathcal L}_{\mathcal U}\) happens to be the infinitesimal generator of the latter semigroup. For instance, the spectrum of \({\mathcal L}_{\mathcal U}\) does not depend on the \(p\in [1,+\infty]\). Another application is a simple criterion for invertibility of the generator in question. The right invertibility of \({\mathcal L}_{\mathcal U}\) is also addressed, and put in connection with the so-called exponential dichotomy property of the original family \(\mathcal U\).
strongly continuous semigroups of operators
scientific article; zbMATH DE number 1557393