For any vector space, one may form the tensor algebra with multiplication being the tensor product. For a Hilbert space $\mathcal{H}$, the analogous construction is the Fock space $$ \mathcal{F}(\mathcal{H})=\bigoplus_{n=0}^\infty\mathcal{H}^{\otimes n}. $$ There exist tensor product maps between individual grades of this space, by which I mean: if $\Psi\in\mathcal{H}^{\otimes n}$, $\Phi\in \mathcal{H}^{\otimes m}$, we have a product vector $\Psi\otimes\Phi\in \mathcal{H}^{\otimes (n+m)}$. If we try to extend the tensor product to the whole Fock space, we may formally define, for any $(\Psi_n)_n, (\Phi_n)_n\in\mathcal{F}(\mathcal{H})$, $$ (\Psi_n)_n \otimes (\Phi_n)_n = \bigg(\sum_{k=0}^n\Psi_k\otimes\Phi_{n-k}\bigg)_n. $$ The problem is that the resulting sequence of vectors isn't necessarily square-summable. Hence we define the mutliplier algebra $$ M(\mathcal{F}(\mathcal{H})) =\bigg\{(\Psi_n)_n\in \mathcal{F}(\mathcal{H})\,\bigg\vert\,\forall(\Phi_n)_n\in\mathcal{F}(\mathcal{H}):\sum_{n=0}^\infty\bigg\Vert\sum_{k + l = n}\Psi_k\otimes\Phi_l\bigg\Vert^2<\infty\bigg\}. $$ My question is whether any more explicit characterization of $M(\mathcal{F}(\mathcal{H}))$ is known. For example, if $\mathcal{H}=\mathbb{C}$, the Fock space is isomorphic to the Hardy space $H^2$ via $(a_n)_n\longmapsto\sum_n a_n z^n$. The multiplier algebra $M(H^2)$, defined as $$ M(H^2)=\{\phi\in H^2\,\vert\, \forall f\in H^2:\phi f\in H^2\}, $$ equals $H^\infty$, the space of bounded analytic functions on the disk. Since the pointwise product of functions corresponds at the level of Taylor coefficients precisely to the tensor product we are trying to define, this is one characterization of $M(\mathcal{F}(\mathbb{C}))$.
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