  We present an algebro-geometric perspective on some generalizations, due to
S. Takagi, of the restriction theorem for multiplier ideals. The first version
of the restriction theorem for multiplier ideals was discovered by Esnault and
Viehweg. In a series of papers S. Takagi has discovered generalizations of the
restriction theorem and some formulas for multiplier ideals that follow from
the restriction theorem. He uses the technique of tight closure and reduction
to positive characteristic. We are able to provide an algebro-geometric proof
of generalizations of his restriction theorem and his subadditivity theorem. We
also prove an adjunction formula for relative canonical divisors of factorizing
resolutions of singularities.
