Abstract: | Let O ? R d be a bounded domain of class C 1,1. Let 0 < ε - 1. In L 2(O;C n ) we consider a positive definite strongly elliptic second-order operator B D,ε with Dirichlet boundary condition. Its coefficients are periodic and depend on x/ε. The principal part of the operator is given in factorized form, and the operator has lower order terms. We study the behavior of the generalized resolvent (B D,ε ? ζQ 0(·/ε))?1 as ε → 0. Here the matrix-valued function Q 0 is periodic, bounded, and positive definite; ζ is a complex-valued parameter. We find approximations of the generalized resolvent in the L 2(O;C n )-operator norm and in the norm of operators acting from L 2(O;C n ) to the Sobolev space H 1(O;C n ) with two-parameter error estimates (depending on ε and ζ). Approximations of the generalized resolvent are applied to the homogenization of the solution of the first initial-boundary value problem for the parabolic equation Q 0(x/ε)? t v ε (x, t) = ?(B D,ε v ε )(x, t). |