Li Guo

Volterra integral operators and generalized Reynolds algebras

We study algebraic structures underlying Volterra integral operators, in particular the operator identities satisfied by such operators. While the operator satisfies the Rota-Baxter identity when the kernel of the operator only depends on the phantom (dummy) variable, we show that when the kernel is more generally separable, the operator satisfies a generalized Reynolds identity which, in its original form, can be tracked back to the famous study of Reynolds in fluid mechanics in the late 19th century. Furthermore a generalized differential operator arises which combined with the generalized Reynolds operator provides an algebraic context, called D-Reynolds algebra, to study Volterra operators and equations. For this purpose, completions and free objects of D-Reynolds algebras are constructed. As applications, linearity of a class of integral equations is established. This is a joint work with Richard Gustavson and Yunnan Li.