The functional $\langle \cdot\rangle_N$ denotes the expectation in the canonical ensamble of particle number $N$. So the grand canonical expectation value $\langle \cdot\rangle$ takes sequences of observables to a real.

We adopt the names of observables in canonical ensamble for the grand canonical ensamble. For example, if the internal energy in the canonical ensamble is defined as $U=\langle H\rangle$, then the grand canonical expectation value of the energy is denoted by $U$ as well and if formed from the sequence of Hamiltonians $H_N$.

Now we also introduce the sequence $\hat N$ which gives us the particle number of each canonical ensamble, i.e. $\langle\hat N_N\rangle_N=N$. Then the expected particle number of the grand canonical ensemble is given by $\langle\hat N\rangle=\sum_{N=0}^\infty w_N\ N$. (Notice that $\hat N$ is sometimes denoted by $N$, which can get a little confusing.)

Grand canonical weight