  In $L_2(\mathbb{R}^d;\mathbb{C}^n)$, consider a self-adjoint matrix second
order elliptic differential operator $\mathcal{B}_\varepsilon$, $0<\varepsilon
\leqslant 1$. The principal part of the operator is given in a factorised form,
the operator contains first and zero order terms. The operator
$\mathcal{B}_\varepsilon$ is positive definite, its coefficients are periodic
and depend on $\mathbf{x}/\varepsilon$. We study the behaviour in the small
period limit of the operator exponential $e^{-\mathcal{B}_\varepsilon t}$,
$t\geqslant 0$. The approximation in the $(L_2\rightarrow L_2)$-operator norm
with error estimate of order $O(\varepsilon ^2)$ 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.

Analysis of PDEs