Splitting of perturbated differential operators with unbounded operator coefficients

We obtain some theorems on splitting of differential operators of the form
$$ \mathcal L=\frac{d}{dt}-A_0-BA_0^\nu\colon\, D(\mathcal L)\subset C(\mathbb R,\mathcal Y)\to C(\mathbb R,\mathcal Y) $$
acting in the Banach space $C(\mathbb R,\mathcal Y)$ of continuous and bounded functions defined on real axis $\mathbb R$ with values in the Banach space $\mathcal Y$. The linear operator $A_0\colon\,D(A_0)\subset\mathcal Y\to\mathcal Y$ is the generating operator of a strongly continuous semigroup of operators and its spectrum does not intersect the imaginary axis $i\mathbb R$. Here $A_0^\nu$, $\nu\in[0,1)$, is a fractional power of $A_0$ and $B\colon\,C(\mathbb R,\mathcal Y)\to C(\mathbb R,\mathcal Y)$ is a bounded linear operator.