Representations of polynomial covariance type commutation relations by linear integral operators with separable kernels in $L_p$

Abstract

Representations of polynomial covariance type commutation relations by linear integral operators on $L_p$ over measures spaces are investigated. Necessary and sufficient conditions for integral operators to satisfy polynomial covariance type commutation relations are obtained in terms of their kernels. For important classes of polynomial covariance commutation relations associated to arbitrary monomials and to affine functions, these conditions on the kernels are specified in terms of the coefficients of the monomials and affine functions. By applying these conditions, examples of integral operators on $L_p$ spaces, with separable kernels representing covariance commutation relations associated to monomials, are constructed for the kernels involving multi-parameter trigonometric functions, polynomials, and Laurent polynomials on bounded intervals. Commutators of these operators are computed and exact conditions for commutativity of these operators in terms of the parameters are obtained.