  We consider a Trotter-type-product formula for approximating the solution of
a linear abstract Cauchy problem (given by a strongly continuous semigroup),
where the underlying Banach space is a product of two spaces. In contrast to
the classical Trotter-product formula, the approximation is given by freezing
subsequently the components of each subspace. After deriving necessary
stability estimates for the approximation, which immediately provide
convergence in the natural strong topology, we investigate convergence in the
operator norm. The main result shows that an almost optimal convergence rate
can be established if the dominant operator generates a holomorphic semigroup
and the off-diagonal coupling operators are bounded.
