Weakly closed semigroups generated by operator-valued functions

The function $P(T)={\sum }_{i=0}^{\mathrm{\infty }}{c}_i{T}^i$ is admissible if $c_i\ge 0$, ${\sum }_{i=0}^{\mathrm{\infty }}{c}_i\le 1$. For any given set of admissible functions $P_1,\dots ,P_k$ there is a unitary operator $T$ of dynamic origin such that the weak closure of its powers is a semigroup generated by the operators $0$, $T$, $P_1(T),\dots ,P_k(T)$, $T^{\ast }$, $P_1(T^{\ast }),\dots ,P_k(T^{\ast })$.