  This paper focuses on representations of contractively embedded invariant
subspaces in several variables. We present a version of the de Branges theorem
for $n$-tuples of multiplication operators by the coordinate functions on
analytic reproducing kernel Hilbert spaces over the unit ball $\mathbb{B}^n$
and the Hardy space over the unit polydics $\mathbb{D}^n$ in $\mathbb{C}^n$.
