Unitary operator - Knowlpedia

Let $H$ and $K$ be complex Hilbert spaces. A unitary operator is a surjective linear map $U:H\to K$ that preserves inner products:

\langle Ux,Uy\rangle_K=\langle x,y\rangle_H \qquad(x,y\in H).

Equivalently, $U$ is bounded and its Hilbert-space adjoint satisfies

U^*U=I_H,\qquad UU^*=I_K.

In that case $U^{-1}=U^*$ and $\lVert Ux\rVert=\lVert x\rVert$ . When $H=K$ , the unitary operators form a group under composition, called the unitary group of $H$ . Surjectivity is essential: an isometric embedding need not be unitary.

Equivalent characterizations

For a bounded linear map $U:H\to K$ , the following are equivalent: $U$ is unitary; $U$ is a surjective isometry; $U^*U=I_H$ and $UU^*=I_K$ ; and $U$ sends some orthonormal basis of $H$ onto an orthonormal basis of $K$ . The single identity $U^*U=I_H$ asserts only that $U$ is an isometry, while $UU^*=I_K$ supplies surjectivity.

Spectral and geometric properties

A unitary operator on $H$ is normal, has operator norm $1$ when $H\neq0$ , and has spectrum contained in the unit circle. Unitary conjugation $T\mapsto U T U^*$ preserves adjoints, products, spectra, operator norms, and singular values. Thus unitary equivalence expresses a change of Hilbert-space coordinates without changing intrinsic operator data.

Examples and non-examples

Multiplication by a measurable function $u$ with $|u|=1$ almost everywhere is unitary on $L^2$ . The bilateral shift on $\ell^2(\mathbb Z)$ is unitary. The unilateral shift on $\ell^2(\mathbb N)$ preserves norms but is not surjective, so it is an isometry rather than a unitary operator.

Conventions and scope

For real Hilbert spaces, the analogous maps are normally called orthogonal operators. In physics, “unitary” may also describe an antiunitary symmetry informally, but an antiunitary map is conjugate-linear and is not unitary under the definition above.

References

John B. Conway, A Course in Functional Analysis, 2nd ed., Springer, 1990. DOI record. Relevant: Chapter II on Hilbert-space operators, adjoints, isometries, and unitaries.