Homogenization of periodic parabolic systems in the $L_2({\mathbb{R}}^d)$-norm with the corrector taken into account

In $L_2({\mathbb{R}}^d;{\mathbb{C}}^n)$, consider a self-adjoint matrix second order elliptic differential operator ${\mathcal{B}}_{\epsilon }$, $0<\varepsilon \leqslant 1$. The principal part of the operator is given in a factorised form, the operator contains first and zero order terms. The operator ${\mathcal{B}}_{\epsilon }$ is positive definite, its coefficients are periodic and depend on $\mathbf{x}\mathrm{/}\epsilon$. We study the behaviour in the small period limit of the operator exponential $e^{-{\mathcal{B}}_{\epsilon }t}$, $t⩾0$. The approximation in the $(L_2\to L_2)$-operator norm with error estimate of order $O({\epsilon }^2)$ is obtained. The corrector is taken into account in this approximation. The results are applied to homogenization of the solutions for the Cauchy problem for parabolic systems.