  We prove that the Neumann, Dirichlet and regularity problems for divergence
form elliptic equations in the half space are well posed in $L_2$ for small
complex $L_\infty$ perturbations of a coefficient matrix which is either real
symmetric, of block form or constant. All matrices are assumed to be
independent of the transversal coordinate. We solve the Neumann, Dirichlet and
regularity problems through a new boundary operator method which makes use of
operators in the functional calculus of an underlaying first order Dirac type
operator. We establish quadratic estimates for this Dirac operator, which
implies that the associated Hardy projection operators are bounded and depend
continuously on the coefficient matrix. We also prove that certain transmission
problems for $k$-forms are well posed for small perturbations of block
matrices.
