Shift operator

# Shift operator

Discussion

Encyclopedia
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, and in particular functional analysis
Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...

, the shift operator or translation operator is an operator that takes a function
to its translation . 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 rôle in harmonic analysis
Harmonic analysis
Harmonic analysis is the branch of mathematics that studies the representation of functions or signals as the superposition of basic waves. It investigates and generalizes the notions of Fourier series and Fourier transforms...

, for example, it appears in the definitions of almost periodic functions, positive definite functions, and convolution
Convolution
In mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation...

. Shifts of sequences (functions of an integer variable) appear in diverse areas such as Hardy space
Hardy space
In complex analysis, the Hardy spaces Hp are certain spaces of holomorphic functions on the unit disk or upper half plane. They were introduced by Frigyes Riesz , who named them after G. H. Hardy, because of the paper...

s, the theory of abelian varieties
Abelian variety
In mathematics, particularly in algebraic geometry, complex analysis and number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions...

, and the theory of symbolic dynamics
Symbolic dynamics
In mathematics, symbolic dynamics is the practice of modeling a topological or smooth dynamical system by a discrete space consisting of infinite sequences of abstract symbols, each of which corresponds to a state of the system, with the dynamics given by the shift operator...

, for which the baker's map
Baker's map
In dynamical systems theory, the baker's map is a chaotic map from the unit square into itself. It is named after a kneading operation that bakers apply to dough: the dough is cut in half, and the two halves are stacked on one-another, and compressed...

is an explicit representation.

### Functions of a real variable

The shift operator takes a function on R to its translation ,

### Sequences

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

and on two-sided infinite sequences by

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

and on two-sided infinite sequences by

### Abelian groups

In general, if is a function on an Abelian group
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...

, and is an element of , the shift operator maps to

## 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
Norm (mathematics)
In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector...

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
Unitary operator
In functional analysis, a branch of mathematics, a unitary operator is a bounded linear operator U : H → H on a Hilbert space H satisfyingU^*U=UU^*=I...

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 is the multiplication operator
Multiplication operator
In operator theory, a multiplication operator is a linear operator T defined on some vector space of functions and whose value at a function φ is given by multiplication by a fixed function f...

by . Therefore the spectrum of is the unit circle.

The one-sided shift acting on is a proper isometry
Isometry
In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...

with range
Range (mathematics)
In mathematics, the range of a function refers to either the codomain or the image of the function, depending upon usage. This ambiguity is illustrated by the function f that maps real numbers to real numbers with f = x^2. Some books say that range of this function is its codomain, the set of all...

equal to all vectors which vanish in the first coordinate. The operator S is a compression
Compression (functional analysis)
In functional analysis, the compression of a linear operator T on a Hilbert space to a subspace K is the operatorP_K T \vert_K : K \rightarrow K...

of T−1, in the sense that

where is the vector in with  =  for and  =  for . This observation is at the heart of the construction of many unitary dilation
Unitary dilation
In operator theory, a dilation of an operator T on a Hilbert space H is an operator on a larger Hilbert space K , whose restriction to H is T....

s of isometries.

The spectrum of S is the unit disk. The shift S is one example of a Fredholm operator
Fredholm operator
In mathematics, a Fredholm operator is an operator that arises in the Fredholm theory of integral equations. It is named in honour of Erik Ivar Fredholm....

; it has Fredholm index −1.

## Generalisation

Jean Delsarte
Jean Delsarte
Jean Frédéric Auguste Delsarte was a French mathematician known for his work in mathematical analysis, in particular, for introducing mean-periodic functions and generalised shift operators. He was one of the founders of the Bourbaki group.-External links:...

introduced the notion of generalised shift operator (also called generalised displacement operator); it was further developed by Boris Levitan
Boris Levitan
Boris Levitan was a mathematician known in particular for his work on almost periodic functions, and Sturm–Liouville operators, especially, on inverse scattering.-Life:...

.

A family of operators acting on a space of functions from a set to is called a family of generalised shift operators if the following properties hold:
1. Associativity: let  = . Then  = .
2. There exists such that is the identity operator.

In this case, the set is called a hypergroup.