  This paper is devoted to morphisms killing weights in a range (as defined by
the first author) and to objects without these weights (as essentially defined
by J. Wildeshaus) in a triangulated category endowed with a weight structure w.
We describe several new criteria for morphisms and objects to satisfy these
conditions. In some of them we use virtual t-truncations and a t-structure
adjacent to w. In the case where the latter exists we prove that a morphism
kills weights $m,...,n$ if and only if it factors through an object without
these weights; we also construct new families of torsion theories and
projective and injective classes.
  As a consequence, we obtain some "weakly functorial decompositions" of
spectra (in the stable homotopy category SH) and a new description of those
morphisms that act trivially on degree zero singular cohomology with
coefficients in every abelian group.
