First the authors restate the definition of quasi-affinity given by J. Apostol [Proc. Am. Math. Soc. 53, 104-106 (1976; Zbl 0323.47017)]: let X,Y be two Banach spaces, an injective, bounded linear operator \(T: X\to Y\) is called a quasi-affinity between X and Y.
Next the authors set up their working hypothesis:
1. let S be a bounded linear operator in X;
2. let V be an injective bounded linear operator such that \([S,V^ 2]=V^ 2\), i.e. V is an injective S-Volterra operator, and 3. let \(A:=V^{-1}\) (with domain \(D(A):=Ran(V))\), let -A be the infinitesimal generator of a uniformly bounded strongly continuous semigroup \(\{U(t)| t\geq 0\}\) in X.
Then for very \(\alpha\in (0,1)\) the bounded operator \(V^{\alpha}\) is defined by means of the theory of fractional powers of closed operators [A. V. Balakrishnan, Pac. J. Math. 10, 419-437 (1960; Zbl 0103.335)], furthermore the fractional powers \(A^{1-\alpha}\) are well defined by an absolutely convergent Balakrishnan integral, and for every \(\alpha\in (0,1)\) the operator \(-A^{1-\alpha}/(1-\alpha)\) is the generator of a strongly continuous semigroup \(\{T_{\alpha}(t)| t\geq 0\}.\)
The authors’ main result states that for very \(\alpha\in (0,1)\), \(t\geq 0\), the strongly continuous semigroup \(\{T_{\alpha}(t)|\) \(t\geq 0\}\) is a quasi-affinity (in X) and \[ S=T_{\alpha}(t)(S+t\cdot V^{\alpha})T_{\alpha}(t)^{-1},\quad (S-t\cdot V^{\alpha})=T_{\alpha}(t)ST_{\alpha}(t)^{-1}, \] i.e. S is a quasi-affine transformation of \(S+t\cdot V^{\alpha}\), and \(S-t\cdot V^{\alpha}\) is quasi-affine transformation of S. Using some earlier results of the first author [e.g.: Spectral theory of Banach space operators, Lect. Notes Math. 1012 (1983; Zbl 0527.47001)] and without the assumption that V can be imbedded into a regular semigroup, the similarity of the operator \(S+\zeta \cdot V^{\alpha}\) and the operator V is shown (\(\zeta\in C\); \(\alpha\in C\), \(Re(\alpha)>1)\). Finally the authors give an alternative proof of these results containing the additional information that the quasi-affinity relations mentioned above are limits of similarity relations between S and bounded operators approximating the operators \(S+t\cdot V^{\alpha}:\)
for \(\epsilon >0\) let \(B_{\epsilon}:=R(\epsilon,-V)=(\epsilon +V)^{- 1}\), for every \(t>0\) and \(\alpha\in (0,1)\) the quasi-affinity relation \(ST_{\alpha}(t)=T_{\alpha}(t)(S+t\cdot V^{\alpha})\) is the limit in the strong operator topology (as \(\epsilon\to 0^+)\) of the similarity relation \[ S \exp[-t\cdot B_{\epsilon}^{1-\alpha} / (1- \alpha)]=\exp[-t\cdot B_{\epsilon}^{1-\alpha} / (1-\alpha)](S+t\cdot B_{\epsilon}^{2-\alpha}V^ 2). \]