I'm currently reading this paper of Marcel Griesemer about the Dirac-Frenkel-approximation method for so-called polarons. It is a mathematical paper and I'm currently confused with a certain notation he establishes:
In Section 6, page 6, he defines the Hilbert-space $\mathcal{H} = L^2(\mathbb{R}^d) \otimes \mathcal{F}$, where $\mathcal{F}$ is the symmetric Fock space over $L^2(\mathbb{R}^d,\mathrm{d} k)$. Then -- everything is fine so far -- he defines $a(f)$ and $a^\ast(f)$ to be the creation and annhilation operators on the Fock space $\mathcal{F}$ with respect to a function $f\in L^2(\mathbb{R}^d,\mathrm{d} k)$.
Here comes the confusing sentence:
It is useful and convenient to extend the notions of creation and annihilation operators to include operators $a^\ast(G)$ and $a(G)$ in $\mathcal{H}$, where the map $k \mapsto G(k)$ takes values in the bounded operators on the particle space $L^2(\mathbb{R}^d)$.
There is no explanation on how exactly these operators are defined, how they act, whatsoever. I would be glad if someone could specify this notion to me.