Shift operator

In mathematics, and in particular functional analysis, the shift operators are examples of linear operators, important for their simplicity and natural occurrence. They are used 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. (There is another usage of shift operator as a translation operator: see for example Sheffer sequence.) In time series analysis, this operator is called the Lag operator. A typical one-sided shift operator takes an infinite sequence of numbers

(a₁, a₂, ...)

to

(0, a₁, a₂, ...).

This operation respects typical convergence conditions, such as absolute convergence of the corresponding infinite series; it therefore gives rise to continuous operators on the standard sequence spaces used in functional analysis, usually with norm 1. Another way to look at it would be in terms of polynomials: the sequences that eventually end in a string

(..., 0, 0, 0, ...)

or, in other words, having only a finite number of non-zero entries, are in a 1-1 correspondence with polynomials in an indeterminate T having a_i as coefficient of Tⁱ. The advantage of this representation is then that the shift operator becomes multiplication by T: this reveals quickly several aspects of its structure. Spaces of polynomials carry numerous topological structures; shift operators can be constructed by extension on corresponding complete spaces. The bilateral shift operators are the related operators in which the sequences are bi-infinite (functions on the integers, rather than just the natural numbers). One can say that the analogue in this case of the polynomial representation is that by Laurent polynomials. The theory of analytic functions is related to that of polynomials, by allowing infinite power series; on the other hand meromorphic functions have Laurent series that terminate in the direction of negative exponents. In the same way, the one-sided and bilateral shifts have rather different properties. This connection with function theory is made more precise in the context of Hardy spaces.

Action on Hilbert spaces

The unilateral and bilateral shifts have a natural action on Hilbert spaces, giving bounded operators S and T on the ℓ^p sequence spaces <math>\ell^2(\mathbb{N})</math> and <math>\ell^2(\mathbb{Z})</math> respectively. The unilateral shift S is a proper isometry with range equal to all vectors which vanish in the first coordinate. The bilateral shift U, on the other hand, is a unitary operator. The operator S is a compression of U, in the sense that

<math>Ux'=Sx \mbox{ for each } x \in \ell^2(\mathbb{N})</math>,

where <math>x'</math> is the vector in <math>\ell^2(\mathbb{Z})</math> with <math>x'_i=x_i</math> for <math>i \geq 0</math> and <math>x'_i=0</math> for <math>i<0</math>. This observation is at the heart of the construction of many unitary dilations of isometries. The spectrum of S is the unit disk while the spectrum of U is the unit circle in the complex plane. The Wold decomposition says that every isometry on a Hilbert space is of the form

<math>S^{\alpha} \oplus U</math>

where S^α is S to the power of some cardinal number α and U is a unitary operator. In turn, the C*-algebra generated by an arbitrary proper isometry is isomorphic to the C*-algebra generated by S. The shift S is one example of a Fredholm operator; it has Fredholm index -1.

See also

• Dilation