Linear integral operators on \(L_p\) spaces representing polynomial covariance type commutation relations

Abstract

In this work, we present methods for constructing representations of polynomial covariance type commutation relations \(AB=BF(A)\) by linear integral operators in Banach spaces \(L_p\). We derive necessary and sufficient conditions on the kernel functions for the integral operators to satisfy the covariance type commutation relation for general polynomials F, as well as for important cases, when F is arbitrary affine or quadratic polynomial, or arbitrary monomial of any degree. Using the obtained general conditions on the kernels, we construct concrete examples of representations of the covariance type commutation relations by integral operators on \(L_p\). Also, we derive useful general reordering formulas for the integral operators representing the covariance type commutation relations, in terms of the kernel functions.