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
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 .
properties
In the following, the additions to the scalar products are omitted, since the argument makes it clear which space is involved.
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
- ,
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 .
Operator norm
The following applies to the operator norm of a unitary operator due to the maintenance of the norm
- .
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
- ,
because it applies
- .
Conversely, if the inverse and adjoint of a linear operator coincide, then this is unitary because it holds
- .
normality
Due to the agreement of the inverse and adjoint, a unitary operator in the case is always normal , that is
- .
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
- .
If vice versa and Hilbert bases of and and is linear, then the unitarity of follows , because one obtains
See also
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)