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We consider the problem of homogenization for non-self-adjoint second-orderelliptic differential operators~$\mathcal{A}^{\varepsilon}$ of divergence formon $L_{2}(\mathbb{R}^{d_{1}}\times\mathbb{T}^{d_{2}})$, where $d_{1}$ ispositive and~$d_{2}$ is non-negative. The~coefficients of theoperator~$\mathcal{A}^{\varepsilon}$ are periodic in the first variable withperiod~$\varepsilon$ and smooth in a certain sense in the second. We show that,as $\varepsilon$ gets small, $(\mathcal{A}^{\varepsilon}-\mu)^{-1}$and~$D_{x_{2}}(\mathcal{A}^{\varepsilon}-\mu)^{-1}$ converge in the operatornorm to, respectively, $(\mathcal{A}^{0}-\mu)^{-1}$and~$D_{x_{2}}(\mathcal{A}^{0}-\mu)^{-1}$, where $\mathcal{A}^{0}$ is anoperator whose coefficients depend only on~$x_{2}$. We also obtain anapproximation for $D_{x_{1}}(\mathcal{A}^{\varepsilon}-\mu)^{-1}$ and find thenext term in the approximation for~$(\mathcal{A}^{\varepsilon}-\mu)^{-1}$.Estimates for the rates of convergence and the rates of approximation areprovided and are sharp with respect to the order.

Homogenization for non-self-adjoint periodic elliptic operators on an infinite cylinder