Bounded Weyl pseudodifferential operators in Fock space

Abstract: We aim at constructing an analog of the Weyl calculus in an infinite dimensional setting, in which the usual configuration and phase spaces are ultimately replaced by infinite dimensional measure spaces, the so-called abstract Wiener spaces. The Hilbert space on which the operators act can be seen as a Fock space or, equivalently, as a space of square integrable functions on the configuration space. The construction is not straightforward and needs to split the configuration space into two factors, of which the first one is finite dimensional. Then one defines, for a convenient symbol $F$, a hybrid calculus, acting on the finite dimensional factor as a Weyl operator and on the other one as an anti-Wick operator, defined thanks to an infinite dimensional Segal-Bargmann transformation. One can establish bounds on the hybrid operators. These bounds enable us to prove the convergence of any sequence of hybrid operators associated with an increasing sequence of finite dimensional factors. Their common limit is the Weyl operator $OP_h^{weyl}(F)$, the analog of Calder\'on-Vaillancourt Theorem being a consequence of the upper mentionned bounds as well.