Unitary Identities

Formula

U^\dagger U=UU^\dagger=I, \qquad U^{-1}=U^\dagger.

Unitary maps preserve inner products and norms:

\langle U\phi|U\psi\rangle=\langle\phi|\psi\rangle.

Products, inverses, tensor products, and direct sums of unitary operators are unitary. Conjugation preserves operator products and commutators:

(UAU^\dagger)(UBU^\dagger)=UABU^\dagger,

[UAU^\dagger,UBU^\dagger]=U[A,B]U^\dagger.

For $H=H^\dagger$ ,

U(t)=e^{-iHt/\hbar}

is unitary. In finite dimension,

U=\sum_j e^{i\theta_j}P_j.

Assumptions and Conventions

$U$ is linear on a complex Hilbert space.
Active transformation uses $A\mapsto UAU^\dagger$ ; a common passive basis convention uses $A_{\mathrm{new}}=U^\dagger A_{\mathrm{old}}U$ .
The finite-dimensional determinant test $|\det U|=1$ is necessary, not sufficient.

Symbols

Symbol	Meaning
$U^\dagger$	adjoint of $U$
$P_j$	spectral projector
$e^{i\theta_j}$	unit-modulus eigenvalue
$H$	self-adjoint generator

Validity and Warnings

Unitary does not mean Hermitian. Operators that are both have eigenvalues $\pm1$ .
In infinite dimension, $U^\dagger U=I$ alone describes an isometry; require surjectivity or also $UU^\dagger=I$ .
A reduced subsystem’s noisy, measured, or postselected dynamics is not generally unitary even when a larger closed system evolves unitarily.
An overall operator phase can become relative under coherent control.
Do not use a unitary polar correction to hide genuine loss or gain in data.

Canonical Treatment

Equivalent tests, active/passive transformations, generators, infinite-dimensional counterexamples, numerical checks, exercises, and references are at Unitary Operators.