In the book "Gaussian Hilbert Spaces" (Svante Janson) the author introduces the Wick product of a finite sequence of $n$ random variables living in a Gaussian Hilbert space $G$ as the orthonormal projection of their product in the $n$-th Wiener chaos, namely:
$$:\xi_1\cdots \xi_n:=\pi_n(\xi_1\cdots\xi_n)$$ where the mapping $\pi_n$ is the orthogonal projection of $L^2(\Omega,\mathcal F,P)\supset G$ into the $n$-th Wiener chaos.
From this definition (it may be trivial actually) but I don't see how to conclude that the $E(:\xi_1\cdots \xi_n:)=0$, other authors use a different approach defining the Wick product recursively for instance here.
Could you please shed some light on this matter?
Thanks in advance!
(I would prefer some explanation that avoids using multiple Ito-Wiener integrals)
As I mentioned in the comments, the easiest way to see this is to notice that $$\pi_n(\xi_1 \dots \xi_n) \in H^{:n:} \perp H^{:0:} = \mathbb{R}$$ so that $$\mathbb{E}[\mathpunct{:} \xi_1 \dots \xi_n \mathpunct{:}] = \langle \mathpunct{:} \xi_1 \dots \xi_n \mathpunct{:}, 1 \rangle_{L^2} = 0.$$