Shift operator

Functions of a real variable The shift operator (where ) takes a function on to its translation , : T^t f(x) = f_t(x) = f(x+t)~. A practical operational calculus representation of the linear operator in terms of the plain derivative {{tmath|\tfrac{d}{dx} }} was introduced by Lagrange, {{Equation box 1 which may be interpreted operationally through its formal Taylor expansion in ; and whose action on the monomial is evident by the binomial theorem, and hence on all series in , and so all functions as above. This, then, is a formal encoding of the Taylor expansion in Heaviside's calculus. The operator thus provides the prototype for Lie's celebrated advective flow for Abelian groups, : \exp\left(t \beta(x) \frac{d}{dx}\right) f(x) = \exp\left(t \frac{d}{dh}\right) F(h) = F(h+t) = f\left(h^{-1}(h(x)+t)\right), where the canonical coordinates (Abel functions) are defined such that :h'(x)\equiv \frac 1 {\beta(x)} ~, \qquad f(x)\equiv F(h(x)). For example, it easily follows that \beta (x)=x yields scaling, : \exp\left(t x \frac{d}{dx}\right) f(x) = f(e^t x) , hence \exp\left(i\pi x \tfrac{d}{dx}\right) f(x) = f(-x) (parity); likewise, \beta (x)=x^2 yields : \exp\left(t x^2 \frac{d}{dx}\right) f(x) = f \left(\frac{x}{1-tx}\right), \beta (x)= \tfrac{1}{x} yields : \exp\left(\frac{t} {x} \frac{d}{dx}\right) f(x) = f \left(\sqrt{x^2+2t} \right) , \beta (x)=e^x yields : \exp\left (t e^x \frac d {dx} \right ) f(x) = f\left (\ln \left (\frac{1}{e^{-x} - t} \right ) \right ) , etc. The initial condition of the flow and the group property completely determine the entire Lie flow, providing a solution to the translation functional equation :f_t(f_\tau (x))=f_{t+\tau} (x) . Sequences The left shift operator acts on one-sided infinite sequence of numbers by : S^*: (a_1, a_2, a_3, \ldots) \mapsto (a_2, a_3, a_4, \ldots) and on two-sided infinite sequences by : T: (a_k)_{k\,=\,-\infty}^\infty \mapsto (a_{k+1})_{k\,=\,-\infty}^\infty. The right shift operator acts on one-sided infinite sequence of numbers by : S: (a_1, a_2, a_3, \ldots) \mapsto (0, a_1, a_2, \ldots) and on two-sided infinite sequences by : T^{-1}:(a_k)_{k\,=\,-\infty}^\infty \mapsto (a_{k-1})_{k\,=\,-\infty}^\infty. The right and left shift operators acting on two-sided infinite sequences are called bilateral shifts. A finite-dimensional analog is given by the shift matrices. Abelian groups In general, as illustrated above, if is a function on an abelian group , and is an element of , the shift operator maps to : F_g(h) = F(h+g). ==Properties of the shift operator==

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. Action on Hilbert spaces 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: \mathcal{F} T^t = M^t \mathcal{F}, where is the multiplication operator by . Therefore, the spectrum of is the unit circle. The one-sided shift acting on is a proper isometry with range equal to all vectors which vanish in the first coordinate. The operator is a compression of , in the sense that T^{-1}y = Sx \text{ for each } x \in \ell^2(\N), where is the vector in with for and for . This observation is at the heart of the construction of many unitary dilations of isometries. The spectrum of is the unit disk. The shift is one example of a Fredholm operator; it has Fredholm index −1. ==Generalization==

Jean Delsarte introduced the notion of generalized shift operator (also called generalized displacement operator); it was further developed by Boris Levitan. A family of operators {{tmath|\{L^x\}_{x \in X} }} acting on a space of functions from a set to is called a family of generalized shift operators if the following properties hold: • Associativity: let (R^y f)(x) = (L^x f)(y). Then L^x R^y = R^y L^x. • There exists in such that is the identity operator. In this case, the set is called a hypergroup. ==See also==