Let \(\mathbb{H}_1,\mathbb{H}_2\) denote complex Hilbert spaces, \(\dim(\mathbb{H}_2) < \infty\), and \(\mathbb{E}\) the space of bounded linear operators from \(\mathbb{H}_1\) to \(\mathbb{H}_2\). The author considers the open unit ball \(\mathbb{B}\) of \(\mathbb{E}\) with its holomorphic invariant metric \(d_{\mathbb{B}}\) such that \(d_{\mathbb{B}}(0, x) = \mathrm{artanh}\Vert x\Vert \) (Carathéodory distance of \(\mathbb{B}\)) along with a \(C_0\)-semigroup \(\{\Phi^t:t\in{\mathbb{R}}_+\}\subset \text{Iso}(d_{\mathbb{B})} := \{\text{holomorphic } d_{\mathbb{B}} \text{-isometries}\}\). with (non-linear) infinitesimal generator \(\Phi'\).
The Möbius transformation is a surjective biholomorphic Carathéodory isometry of the unit ball. Two \(C_0\)-semigroups \(\Phi^t\) and \(\Phi^t\) are Möbius equivalent if \(\Psi^t=\Theta \Phi^t \Theta^{-1}\).
The main result of the paper under review is the following.
Let \(\Psi^t\) be a \(C_0\)-semigroup not Möbius equivalent to a \(C_0\)-semigroup of linear isometries. Then there is a Möbius equivalent \(C_0\)-semigroup \(\Phi^t\) to \(\Psi^t\) of the form \[ \Phi^t(X)=E+W^t(X-E)\left[\int\limits_0^t S^{t-h}b\ast W^h(X-E)\, dh+S^t\right]^{-1} \text{ for all }X\in \mathbb{B} \] for some \(b\in \mathbb{E}\), where \(E\in\partial \mathbb{B}\) is a common fixed point of \(\Phi^t\), \(E\) is a partial isometry, and \(W^t\), \(S^t\) are suitable \(C_0\)-semigroups.
The structure of the semigroups \(W^t\), \(S^t\) is also examined.