Perturbation theorems for \(\alpha\)-times integrated semigroups
Under relative smallness conditions for the perturbing operator \(B\), perturbation results are proved for the generator \((A,D(A))\) of an \(\alpha\)-times integrated semigroup on a Banach space \(X\). More precisely, it is shown that if the estimate \[ \| B(\lambda-A)^{-1}\| \leq M<1\quad\text{or}\quad\| (\lambda-A)^{-1}Bx\| \leq M\| x\| \hskip5pt\text{for \(x\in D(B)\)} \] holds for \(\lambda\in\mathbb{C}\) on a vertical line, then (an extension of) \(A+B\) generates a \(\beta\)-times integrated semigroup, where \(\beta>\alpha+1\) if the \(\alpha\)-times integrated semigroup generated by \(A\) is exponentially bounded, or \(\beta>\alpha+2\) in the general case. For exponentially bounded \(\alpha\)-times integrated semigroups, the result can be sharpened if the Banach space has Fourier type \(p\in[1,2]\). Namely, in this case it suffices to take \(\beta>1+{{1}\over{p}}\). In particular, when \(A\) generates a \(C_0\)-semigroup and \(X\) is \(B\)-convex, the results yield that \(A+B\) (again with appropriate domain, depending on which estimate above is satisfied) is a generator of a once-integrated semigroup. As an illustration, Schrödinger operators and higher order differential operators are considered.
scientific article; zbMATH DE number 2051594