Quasinilpotent variant of Friedrichs' method in the theory of similarity of linear operators (Q799918)

Let X denote a complex Banach space. A closed linear operator A on X is called non-quasianalytic, if it has a representation \(A=A_ 1+iA_ 2\), \(D(A)\subset D(A_ 1)\cap D(A_ 2)\), where \(iA_ 1\), \(iA_ 2\), are generators of strongly continuous groups of operators \(\{T_ 1(t)\}\), \(\{T_ 2(t)\}\), \(t\geq 0\), which commute, and \(\int (\log \| T_ k(t)\| /(1+t^ 2))dt\) is finite, \(k=1,2\). The author asserts that, under certain conditions, if A is a non-quasianalytic operator, and if B is a linear operator with decomposition \(B=\sum_{n}B_ n,\) then the operators \(A+B\) and A are similar. (The theorem is obtained in the Banach algebra setting.)