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In this paper we introduce and study a two-parameter family of integraloperators on the Fock space $F^2(C)$. We determine exactly when these operatorsare bounded and when they are unitary. We show that, under the Bargmanntransform, these operators include the classical linear canonical transforms asspecial cases. As an application, we obtain a new unitary projectiverepresentation for the special linear group $SL(2,R)$ on the Fock space.

Canonical integral operators on the Fock space