Description of the operator dual to the multiplication operator in Fock space (Q1945179)

Let \(n\) be a natural number and \(H\left( \mathbb{C}^{n}\right) \) denote the space of all entire functions on \(\mathbb{C}^{n}\). Recall that the Fock space \(F_{n}\) is given by \[ F_{n}=\left\{ f\in H\left( \mathbb{C}^{n}\right) :\int_{\mathbb{C}^{n} }\mathrm{e}^{-\left| z\right| ^{2}}\left| f\left( z\right) \right| ^{2}\mathrm{d}\mu_{n}\left( z\right) <\infty\right\}, \] where \(\mu_{n}\) is the Lebesgue measure on \(\mathbb{C}^{n}\). Actually, \(F_{n}\) is a Hilbert space with respect to the inner product \[ \left( f|g\right) _{F_{n}}=\frac{1}{\pi^{n}}\int_{\mathbb{C}^{n}} \mathrm{e}^{-\left| z\right| ^{2}}f\left( z\right) \overline {g\left( z\right) }\,\mathrm{d}\mu_{n}\left( z\right) . \] The author of this note describes the operator dual to the operator multiplying elements of \(F_{n}\) by an entire function \(\varphi\) satisfying the condition \[ \varphi\left( .\right) \mathrm{e}^{\left\langle .,\overline{\zeta }\right\rangle }\in F_{n}\text{ for all }\zeta\in\mathbb{C}^{n}. \] His approach is based upon the theory of orthosimilar expansion systems. The main result of the note is then used to prove the uniqueness a Fischer decomposition of certain functions in \(F_{n}\).