Unit operator

In analyzes functional, a unit operator is a linear Opérateur $U$ of a Espace of Hilbert satisfying the conditions:

$U^*U=UU^*=I$

where $U^*$ is the assistant of $U$ , and $I$ the operator identity. This property is equivalent to:

$U$ is an operator with dense field, and
$U$ preserves the scalar Produit < , > on the space of Hilbert. In other words, for all vectors $x$ and $y$ of the space of Hilbert,

\ langle Ux, Uy \ rangle = \ langle X, there \ rangle.

The fact that $U$ preserves the scalar product implies that $U$ is a Isométrie (and thus a linear Opérateur). The fact that $U$ has a dense field ensures that are opposite U ^-1 is limited. It is clear that U ^-1 = U *.

Consequently, the unit operators seem isomorphisms space of Hilbert, i.e. they preserves the structure and topology of it.

Examples

the function identity is, in a commonplace way, a unit operator.

In the vector Space C of the complex numbers, the multiplication by a complex number of module 1 (c.a.d a number of the form E ^{I θ} for θ ∈ R ), is a unit operator. The value of θ modulo 2 $\ pi$ does not affect the result of the multiplication, and consequently the unit operators of C are parameterized by a circle. The group corresponding, whose whole is the circle unit, is called U (1).

more generally, the unit matrices are very exactly the unit operators for spaces of Hilbert of finished size; consequently the concept of operator unit is a generalization of the concept of matrix unit. The orthogonal matrices are a particular case of the unit matrices, for which all the coefficients are real. They are the unit operators of R ^N.

the operator of Fourier (c.a.d the operator which carries out a Transformation of Fourier) is a unit operator (with an adequate standardization). It is a consequence of the Théorème of Parseval.

Properties

the spectrum of a unit operator U is the circle unit. In other words, for any complex number λ spectrum, |λ |=1. It is a consequence of the spectral Théorème for the normal operators.

Random links: