  We define and study cohomological tensor functors from the category $T_n$ of
finite-dimensional representations of the supergroup $Gl(n|n)$ into $T_{n-r}$
for $0 <r \leq n$. In the case $DS: T_n \to T_{n-1}$ we prove a formula $DS(L)
= \bigoplus \Pi^{n_i} L_i$ for the image of an arbitrary irreducible
representation. In particular $DS(L)$ is semisimple and multiplicity free. We
derive a few applications of this theorem such as the degeneration of certain
spectral sequences and a formula for the modified superdimension of an
irreducible representation.
