Commutators on Fock spaces

Abstract

Given a weighted ℓ2 space with weights associated with an entire function, we consider pairs of weighted shift operators, whose commutators are diagonal operators, when considered as operators over a general Fock space. We establish a calculus for the algebra of these commutators and apply it to the general case of Gelfond–Leontiev derivatives. This general class of operators includes many known examples, such as classic fractional derivatives and Dunkl operators. This allows us to establish a general framework, which goes beyond the classic Weyl–Heisenberg algebra. Concrete examples for its application are provided.