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Suppose that $S_t:\ X\to X,\ t\ge 0,$ is a $C_0$-semigroup on the Banach space $X$. Consider the covariant operator-valued measure $\mathcal M$ on the half-axis ${\mathbb R}_+$ taking values in operators on $X$ and satisfying the property
$$
S_t\circ {\mathcal M}(B)={\mathcal M}(B+t),\ t\ge 0,
$$
for all measurable subsets $B \subset \mathbb R_+$.
Theorem. Suppose that ${\mathcal M}$ is semi-absolutely continuous in the sense
$$
\lim \limits _{t\to +0}\overline {\mathcal M}([T,T+t))=0,\ \forall T\ge 0,
$$
where
$$
\overline{\mathcal M}(E) = \sup\Bigl\{\Bigl\| \sum_{i=1}^n {\mathcal M}(E_i)x_i\Bigr\|: x_j\in X,\|x_j\|\le 1, E\supset E_j\subset {\mathbb R}, E_j\mbox{ are disjoint}\Bigr\}
$$
for all measurable $E\subset {\mathbb R}_+$. Then, the solution to the integral equation
$$
\breve S_t=S_t+\int \limits _0^t{\mathcal M}(ds)\circ \breve S_{t-s}
$$
produce the $C_0$-semigroup $\breve S={\breve S_t,\ t\ge 0}$. The generator of $\breve S$ acts as
$$
x\to \lim\limits _{t\to +0}\frac {S_t-I+{\mathcal M}([0,t))}{t}x
$$
with the domain consisting of exactly those $x$ for which the strong limit exists.
Meaningful examples of covariant measures can be obtained from the orbits of unitary groups in infinite-dimensional space [1].
This work was funded by Russian Federation represented by the Ministry of Science
and Higher Education of the Russian Federation (Grant No. 075-15-2020-788) and
performed at the Steklov Mathematical Institute of the Russian Academy of Sciences.
Joint work with E.L. Baitenov.
[1] G. G. Amosov, A. S. Mokeev, A. N. Pechen, Noncommutative graphs based on finite-infinite system couplings: Quantum error correction for a qubit coupled to a coherent field, Phys. Rev. A, 103:4 (2021), 042407, 17 pp.
