The shift operator T t (where ) takes a function f on to its translation ft,
A practical operational calculus representation of the linear operator T t in terms of the plain derivative was introduced by Lagrange,
which may be interpreted operationally through its formal Taylor expansion in t; and whose action on the monomial xn is evident by the binomial theorem, and hence on all series inx, and so all functions f(x) as above.[3] This, then, is a formal encoding of the Taylor expansion in Heaviside's calculus.
The initial condition of the flow and the group property completely determine the entire Lie flow, providing a solution to the translation functional equation[6]
The shift operator acting on real- or complex-valued functions or sequences is a linear operator which preserves most of the standard norms which appear in functional analysis. Therefore, it is usually a continuous operator with norm one.
The shift operator acting on two-sided sequences is a unitary operator on The shift operator acting on functions of a real variable is a unitary operator on
In both cases, the (left) shift operator satisfies the following commutation relation with the Fourier transform:
where M t is the multiplication operator by exp(itx). Therefore, the spectrum of T t is the unit circle.
The one-sided shift S acting on is a proper isometry with range equal to all vectors which vanish in the first coordinate. The operator S is a compression of T−1, in the sense that
where y is the vector in with yi = xi for i ≥ 0 and yi = 0 for i < 0. This observation is at the heart of the construction of many unitary dilations of isometries.
Jean Delsarte introduced the notion of generalized shift operator (also called generalized displacement operator); it was further developed by Boris Levitan.[2][8][9]
A family of operators acting on a space Φ of functions from a set X to is called a family of generalized shift operators if the following properties hold:
^Jordan, Charles, (1939/1965). Calculus of Finite Differences, (AMS Chelsea Publishing), ISBN978-0828400336 .
^M Hamermesh (1989), Group Theory and Its Application to Physical Problems
(Dover Books on Physics), Hamermesh ISBM 978-0486661810, Ch 8-6, pp 294-5,
online.
^p 75 of Georg Scheffers (1891): Sophus Lie, Vorlesungen Ueber Differentialgleichungen Mit Bekannten Infinitesimalen Transformationen, Teubner, Leipzig, 1891. ISBN978-3743343078online
^ abAczel, J (2006), Lectures on Functional Equations and Their Applications (Dover Books on Mathematics, 2006), Ch. 6, ISBN978-0486445236 .
^"A one-parameter continuous group is equivalent to a group of translations". M Hamermesh, ibid.