Unitary operator

In mathematics, a unitary operator is a bijective linear operator between two Hilbert spaces that receives the scalar product . Unitary operators are special orthogonal or unitary mappings and always norm-preserving , distance- preserving , bounded and, if both Hilbert spaces are equal, normal . The inverse operator of a unitary operator is equal to its adjoint operator . The eigenvalues ​​of a unitary operator in a Hilbert space all have the absolute value one. Unitary operators between finite-dimensional vector spaces of the same dimension can be represented by unitary matrices , depending on the choice of an orthonormal basis . Important examples of unitary operators between infinite-dimensional function spaces are the Fourier transform and the time evolution operators of quantum mechanics .

definition

A unitary operator is a bijective linear operator between two Hilbert spaces and such that ${\displaystyle T\colon V\to W}$ ${\displaystyle (V,\langle \cdot ,\cdot \rangle _{V})}$ ${\displaystyle (W,\langle \cdot ,\cdot \rangle _{W})}$

${\displaystyle \langle Tu,Tv\rangle _{W}=\langle u,v\rangle _{V}}$

holds for all vectors . A unitary operator is therefore an isomorphism between two Hilbert spaces that receives the scalar product . A unitary operator between two real Hilbert spaces is sometimes called an orthogonal operator . ${\displaystyle u,v\in V}$

properties

In the following, the additions to the scalar products are omitted, since the argument makes it clear which space is involved. ${\displaystyle V,W}$

Basic characteristics

Every unitary operator represents a unitary mapping (in the real case orthogonal mapping ). The linearity therefore already follows from the conservation of the scalar product and therefore does not have to be required separately. A unitary operator still receives the scalar product norm of a vector, that is, it holds

${\displaystyle \|Tv\|={\sqrt {\langle Tv,Tv\rangle }}={\sqrt {\langle v,v\rangle }}=\|v\|}$,

and thus also the distance between two vectors. The figure thus represents an isometry and the two spaces and are therefore isometrically isomorphic . The eigenvalues ​​of a unitary operator all have the amount one. More generally, the spectrum of a unitary operator lies in the edge of the unit circle . ${\displaystyle T}$ ${\displaystyle V}$ ${\displaystyle W}$ ${\displaystyle T\colon V\to V}$

Operator norm

The following applies to the operator norm of a unitary operator due to the maintenance of the norm ${\displaystyle T}$

${\displaystyle \|T\|=\sup _{\|v\|=1}\|Tv\|=\sup _{\|v\|=1}\|v\|=1}$.

A unitary operator is therefore always bounded and therefore continuous .

Inverse

The inverse operator of a unitary operator is equal to its adjoint operator , so ${\displaystyle T^{-1}}$ ${\displaystyle T}$ ${\displaystyle T^{\ast }}$

${\displaystyle T^{-1}=T^{\ast }}$,

because it applies

${\displaystyle \langle u,T^{\ast }v\rangle =\langle Tu,v\rangle =\langle Tu,TT^{-1}v\rangle =\langle u,T^{-1}v\rangle }$.

Conversely, if the inverse and adjoint of a linear operator coincide, then this is unitary because it holds

${\displaystyle \langle Tu,Tv\rangle =\langle u,T^{\ast }Tv\rangle =\langle u,T^{-1}Tv\rangle =\langle u,v\rangle }$.

normality

Due to the agreement of the inverse and adjoint, a unitary operator in the case is always normal , that is ${\displaystyle V=W}$

${\displaystyle T^{\ast }T=TT^{\ast }=I}$.

The spectral theorem applies to unitary operators on complex Hilbert spaces and self-adjoint unitary operators on real Hilbert spaces .

Base transformation

Is a unitary operator and is a Hilbert basis (a complete orthonormal system) of , then is a Hilbert basis of , because it holds ${\displaystyle T}$ ${\displaystyle (v_{i})_{i\in I}}$ ${\displaystyle V}$ ${\displaystyle (Tv_{i})_{i\in I}}$ ${\displaystyle W}$

${\displaystyle \langle Tv_{i},Tv_{j}\rangle =\langle v_{i},v_{j}\rangle =\delta _{ij}}$.

If vice versa and Hilbert bases of and and is linear, then the unitarity of follows , because one obtains ${\displaystyle (v_{i})_{i\in I}}$ ${\displaystyle (Tv_{i})_{i\in I}}$ ${\displaystyle V}$ ${\displaystyle W}$ ${\displaystyle T}$ ${\displaystyle T}$

${\displaystyle {\begin{aligned}\langle Tu,Tv\rangle &={\big \langle }T{\big (}{\textstyle \sum _{i}}\lambda _{i}v_{i}{\big )},T{\big (}{\textstyle \sum _{j}}\mu _{j}v_{j}{\big )}{\big \rangle }={\big \langle }{\textstyle \sum _{i}}\lambda _{i}Tv_{i},{\textstyle \sum _{j}}\mu _{j}Tv_{j}{\big \rangle }={\textstyle \sum _{i}}{\textstyle \sum _{j}}\lambda _{i}{\bar {\mu }}_{j}{\big \langle }Tv_{i},Tv_{j}{\big \rangle }=\\&={\textstyle \sum _{i}}{\textstyle \sum _{j}}\lambda _{i}{\bar {\mu }}_{j}\delta _{ij}={\textstyle \sum _{i}}{\textstyle \sum _{j}}\lambda _{i}{\bar {\mu }}_{j}\langle v_{i},v_{j}\rangle ={\big \langle }{\textstyle \sum _{i}}\lambda _{i}v_{i},{\textstyle \sum _{j}}\mu _{j}v_{j}{\big \rangle }=\langle u,v\rangle .\end{aligned}}}$

See also

• Hilbert-Schmidt operator
• Hilbert space representation

literature

• Hans Wilhelm Alt: Linear Functional Analysis: An Application-Oriented Introduction . 5th edition. Springer, 2008, ISBN 3-540-34186-2 .
• Dirk Werner: Functional Analysis . 5th edition. Springer, 2005, ISBN 3-540-21381-3 .

Web links

• VI Sobolev: Unitary operator . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
• Eric W. Weisstein : Unitary . In: MathWorld (English).
• asteroid: Unitary . In: PlanetMath . (English)