  The problem of block diagonalization for diagonally dominant symmetric block
operator matrices with self-adjoint diagonal entries is considered. We show
that a reasonable block diagonalization with respect to a reducing graph
subspace requires a related skew-symmetric operator to be a strong solution to
the associated Riccati equation. Under mild additional regularity conditions,
we also establish that this skew-symmetric operator is a strong solution to the
Riccati equation if and only if the graph subspace is reducing for the given
operator matrix. These regularity conditions are shown to be automatically
fulfilled whenever the corresponding relative bound of the off-diagonal part is
sufficiently small. This extends the results by Albeverio, Makarov, and
Motovilov in [Canad. J. Math. Vol. \textbf{55}, 2003, 449--503], where the
off-diagonal part is required to be bounded.
