Abstract
We give a detailed analysis of the extended Fock space construction, a representation of the Heisenberg algebra (the three-dimensional algebra 
with commutators 
and 
) in a separable Hilbert space \(\mathcal {H}\), such that 
and the set, formed by finite linear combinations of elements from the linear subsets 
, \(n \in \mathbb {N}\), is dense in \(\mathcal {H}\). A special attention is paid to the case when 
is formally adjoint to 
. The results obtained are illustrated on the spaces of polyanalytic functions (those satisfying the equation \(\frac {\partial ^n }{\partial \overline {z}^n}f = 0\)), being either the poly-Bergman spaces on the unit disk, or poly-Fock spaces on the complex plane.
This work was partially supported by CONACYT grants 280732 and FORDECYT-PRONACES/61517/2020238630, Mexico.
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Vasilevski, N. (2023). Extended Fock Space Formalism and Polyanalytic Functions. In: Alpay, D., Behrndt, J., Colombo, F., Sabadini, I., Struppa, D.C. (eds) Recent Developments in Operator Theory, Mathematical Physics and Complex Analysis. Operator Theory: Advances and Applications, vol 290. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-21460-8_10
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