Unilateral shift operator

In operator theory, the unilateral shift is an operator on a Hilbert space. It is often studied in two main representations: as an operator on the sequence space , or as a multiplication operator on a Hardy space. Its properties, particularly its invariant subspaces, are well-understood and serve as a model for more general theories.

Definition

Let be the Hilbert space of square-summable sequences of complex numbers, i.e., The unilateral shift is the linear operator defined by: This operator is also called the forward shift.

With respect to the standard orthonormal basis for , where is the sequence with a 1 in the n-th position and 0 elsewhere, the action of is . Its matrix representation is:This is a Toeplitz operator whose symbol is the function . It can be regarded as an infinite-dimensional lower shift matrix.

Properties

Adjoint operator

The adjoint of the unilateral shift, denoted , is the backward shift. It acts on as: The matrix representation of is the conjugate transpose of the matrix for : It can be regarded as an infinite-dimensional upper shift matrix.

Basic properties

are both continuous but not compact.
.
make up a pair of unitary equivalence between and the set of -sequences whose first element is zero.

The resolvent operator has matrix representationwhich is bounded iff . Similarly, .

For any with ,where is the real part.

Spectral theory

The spectral properties of differ significantly from those of :

(since ).
The point spectrum is the entire open unit disk . For any , the corresponding eigenvector is the geometric sequence .
The approximate point spectrum is the entire closed unit disk . To show this, it remains to show , which can be proven by a similar construction as before, using .

Hardy space model

The unilateral shift can be studied using complex analysis.

Define the Hardy space as the Hilbert space of analytic functions on the open unit disk for which the sequence of coefficients is in .

Define the multiplication operator on : then and are unitarily equivalent via the unitary map defined bywhich gives . Using this unitary equivalence, it is common in the literature to use to denote and to treat as the primary setting for the unilateral shift.

Commutant

The commutant of an operator , denoted , is the algebra of all bounded operators that commute with . The commutant of the unilateral shift is the algebra of multiplication operators on by bounded analytic functions.Here, is the space of bounded analytic functions on , and .

Cyclic vectors

A vector is a cyclic vector for an operator if the linear span of its orbit is dense in the space. We have:

For the unilateral shift on , the cyclic vectors are the outer functions.
A function that has a zero in the open unit disk is not a cyclic vector. This is because every function in the span of its orbit will also be zero at that point, so the subspace cannot be dense.
A function that is bounded away from zero (i.e., ) is a cyclic vector.
A function , that is in the open unit disk is nonzero but , may or may not be cyclic. For example, is a cyclic vector.

The cyclic vectors are precisely the outer functions.

Lattice of invariant subspaces

The -invariant subspaces of are completely characterized analytically. Specifically, they are precisely where is an inner function.

The -invariant subspaces make up a lattice of subspaces. The two lattice operators, join and meet, correspond to operations on inner functions.

Given two invariant subspaces , we have iff .

References