Shift operator

In mathematics, and in particular functional analysis, the shift operator, also known as the translation operator, is an operator that takes a function x ↦ f(x) to its translation x ↦ f(x + a). In time series analysis, the shift operator is called the lag operator.

Shift operators are examples of linear operators, important for their simplicity and natural occurrence. The shift operator action on functions of a real variable plays an important role in harmonic analysis, for example, it appears in the definitions of almost periodic functions, positive-definite functions, derivatives, and convolution. Shifts of sequences (functions of an integer variable) appear in diverse areas such as Hardy spaces, the theory of abelian varieties, and the theory of symbolic dynamics, for which the baker's map is an explicit representation. The notion of triangulated category is a categorified analogue of the shift operator.

Definition

Functions of a real variable

The shift operator Template:Mvar (where Template:Tmath) takes a function Template:Mvar on Template:Tmath to its translation Template:Mvar,

$T^{t}f(x)=f_t(x)=f(x+t).$

A practical operational calculus representation of the linear operator Template:Mvar in terms of the plain derivative Template:Tmath was introduced by Lagrange,

which may be interpreted operationally through its formal Taylor expansion in Template:Mvar; and whose action on the monomial Template:Mvar is evident by the binomial theorem, and hence on all series in Template:Mvar, and so all functions f(x) 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,

$\exp (t\beta (x)\frac{d}{dx})f(x)=\exp \left(t\frac{d}{dh}\right)F(h)=F(h+t)=f(h^{-1}(h(x)+t)),$

where the canonical coordinates Template:Mvar (Abel functions) are defined such that

$h^{\prime }(x)\equiv \frac{1}{\beta (x)},f(x)\equiv F(h(x)).$

For example, it easily follows that $\beta (x)=x$ yields scaling,

$\exp \left(tx\frac{d}{dx}\right)f(x)=f(e^{t}x),$

hence $\exp \left(i\pi x\frac{d}{dx}\right)f(x)=f(-x)$ (parity); likewise, $\beta (x)=x^2$ yields

$\exp \left(t{x}^2\frac{d}{dx}\right)f(x)=f\left(\frac{x}{1-tx}\right),$

$\beta (x)=\frac{1}{x}$ yields

$\exp \left(\frac{t}{x}\frac{d}{dx}\right)f(x)=f\left(\sqrt{x^2+2t}\right),$

$\beta (x)=e^x$ yields

$\exp \left(t{e}^x\frac{d}{dx}\right)f(x)=f(\ln \left(\frac{1}{e^{-x}-t}\right)),$

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

$S^{*}:(a_1,a_2,a_3,\dots )\mapsto (a_2,a_3,a_4,\dots )$

and on two-sided infinite sequences by

$T:(a_k{)}_{k=-\mathrm{\infty }}^{\mathrm{\infty }}\mapsto (a_{k+1}{)}_{k=-\mathrm{\infty }}^{\mathrm{\infty }}.$

The right shift operator acts on one-sided infinite sequence of numbers by

$S:(a_1,a_2,a_3,\dots )\mapsto (0,a_1,a_2,\dots )$

and on two-sided infinite sequences by

$T^{-1}:(a_k{)}_{k=-\mathrm{\infty }}^{\mathrm{\infty }}\mapsto (a_{k-1}{)}_{k=-\mathrm{\infty }}^{\mathrm{\infty }}.$

The right and left shift operators acting on two-sided infinite sequences are called bilateral shifts.

Abelian groups

In general, as illustrated above, if Template:Mvar is a function on an abelian group Template:Mvar, and Template:Mvar is an element of Template:Mvar, the shift operator Template:Mvar maps F 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 Template:Tmath The shift operator acting on functions of a real variable is a unitary operator on Template:Tmath

In both cases, the (left) shift operator satisfies the following commutation relation with the Fourier transform: $${ℱ}{T}^t=M^t{ℱ},$$ where Template:Mvar is the multiplication operator by exp(itx). Therefore, the spectrum of Template:Mvar is the unit circle.

The one-sided shift Template:Mvar acting on Template:Tmath is a proper isometry with range equal to all vectors which vanish in the first coordinate. The operator Template:Mvar is a compression of TTemplate:I sup, in the sense that $$T^{-1}y=Sx\text{ for each }x\in {\ell }^2(\mathbb{\mathbb{N}}),$$ where Template:Mvar is the vector in Template:Tmath with y_i = x_i for i ≥ 0 and y_i = 0 for i < 0. This observation is at the heart of the construction of many unitary dilations of isometries.

The spectrum of Template:Mvar is the unit disk. The shift Template:Mvar 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 Template:Tmath acting on a space Φ of functions from a set Template:Mvar to Template:Tmath is called a family of generalized shift operators if the following properties hold:

1. Associativity: let $(R^{y}f)(x)=(L^{x}f)(y).$ Then $L^x{R}^y=R^y{L}^x.$
2. There exists Template:Mvar in Template:Mvar such that Template:Mvar is the identity operator.

In this case, the set Template:Mvar is called a hypergroup.

See also

• Arithmetic shift
• Logical shift
• Clock and shift matrices
• Finite difference
• Translation operator (quantum mechanics)

Bibliography

• Marvin Rosenblum and James Rovnyak, Hardy Classes and Operator Theory, (1985) Oxford University Press.