On perturbations of semigroups based upon operator-valued measures

3 Jul 2021, 17:35


Grigorii Amosov (Steklov Mathematical Institute of RAS)


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.

Primary author

Grigorii Amosov (Steklov Mathematical Institute of RAS)

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