  In $L_2({\mathbb R}^d;{\mathbb C}^n)$, we study a selfadjoint strongly
elliptic operator $A_\varepsilon$ of order $2p$ given by the expression
$b({\mathbf D})^* g({\mathbf x}/\varepsilon) b({\mathbf D})$, $\varepsilon >0$.
Here $g({\mathbf x})$ is a bounded and positive definite $(m\times
m)$-matrix-valued function in ${\mathbb R}^d$; it is assumed that $g({\mathbf
x})$ is periodic with respect to some lattice. Next, $b({\mathbf
D})=\sum_{|\alpha|=p}^d b_\alpha {\mathbf D}^\alpha$ is a differential operator
of order $p$ with constant coefficients; $b_\alpha$ are constant $(m\times
n)$-matrices. It is assumed that $m\ge n$ and that the symbol $b({\boldsymbol
\xi})$ has maximal rank. For the resolvent $(A_\varepsilon - \zeta I)^{-1}$
with $\zeta \in {\mathbb C} \setminus [0,\infty)$, we obtain approximations in
the norm of operators in $L_2({\mathbb R}^d;{\mathbb C}^n)$ and in the norm of
operators acting from $L_2({\mathbb R}^d;{\mathbb C}^n)$ to the Sobolev space
$H^p({\mathbb R}^d;{\mathbb C}^n)$, with error estimates depending on
$\varepsilon$ and $\zeta$.
