  We study the Poincar\'e series of the quantum spaces associated to a Hecke
operator, i.e., a Yang-Baxter operator satisfying the equation $(x+1)(x-q)=0$.
The Poincar\'e series of the corresponding matrix bialgebra is also considered.
Using an old result on Poly\'a frequency sequence, we show that the Poincar\'e
series of quantum spaces are always rational functions having negative roots
and positive poles. In particular, we show that the rank of an even Hecke
operator should be rational functions having negative roots and positive poles.
In particular, we show that the rank of an even Hecke operator should be
greater than the dimension of the vector space it is acting on.
