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In $L_2(\mathbb{R}^d;\mathbb{C}^n)$, we consider a matrix elliptic secondorder differential operator $B_\varepsilon >0$. Coefficients of the operator$B_\varepsilon$ are periodic with respect to some lattice in $\mathbb{R}^d$ anddepend on $\mathbf{x}/\varepsilon$. We study the quantitative homogenizationfor the solutions of the hyperbolic system $\partial _t^2\mathbf{u}_\varepsilon=-B_\varepsilon\mathbf{u}_\varepsilon$. In operator terms, we are interested inapproximations of the operators $\cos (tB_\varepsilon ^{1/2})$ and$B_\varepsilon ^{-1/2}\sin (tB_\varepsilon ^{1/2})$ in suitable operator norms.Approximations for the resolvent $B_\varepsilon ^{-1}$ have been alreadyobtained by T.~A.~Suslina. So, we rewrite hyperbolic equation as a system forthe vector with components $\mathbf{u}_\varepsilon $ and $\partial_t\mathbf{u}_\varepsilon$, and consider the corresponding unitary group. Forthis group, we adapt the proof of the Trotter-Kato theorem by introduction ofsome correction term and derive hyperbolic results from elliptic ones.

Variations on the theme of the Trotter-Kato theorem for homogenization of periodic hyperbolic systems