Shift Operator - Generalisation

Generalisation

Jean Delsarte introduced the notion of generalised shift operator (also called generalised displacement operator); it was further developed by Boris Levitan.

A family of operators {Lx}_{x ∈ X} acting on a space Φ of functions from a set X to C is called a family of generalised shift operators if the following properties hold:

1. Associativity: let (Ryf)(x) = (Lxf)(y). Then LxRy = RyLx.
2. There exists e ∈ X such that Le is the identity operator.

In this case, the set X is called a hypergroup.