In functional analysis, a unitary operator is a bounded linear operator $U$ on a Hilbert space satisfying

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

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

• $U$ is a surjective isometry

• $U$ is surjective and preserves the inner product <  ,  > on the Hilbert space, so that for all vectors $x$ and $y$ in the Hilbert space,

$\backslash langle\; Ux,\; Uy\; \backslash rangle\; =\; \backslash langle\; x,\; y\; \backslash rangle.$

Thus, unitary operators are just isomorphisms between Hilbert spaces, i.e., they preserve the structure (in this case, the linear space structure, the inner product, and hence the topology) of the spaces.

Examples

• The identity function is trivially a unitary operator.

• On the vector space C of complex numbers, multiplication by a number of absolute value 1, that is, a number of the form e^{i θ} for θ ∈ R, is a unitary operator. θ is referred to as a phase, and this multiplication is referred to as multiplication by a phase. Notice that the value of θ modulo 2$\backslash pi$ does not affect the result of the multiplication, and so the independent unitary operators on C are parametrized by a circle. The corresponding group, which as a set is the circle, is called U(1).

• More generally, unitary matrices are precisely the unitary operators on finite-dimensional Hilbert spaces, so the notion of a unitary operator is a generalisation of the notion of a unitary matrix. Orthogonal matrices are the special case of unitary matrices in which all entries are real. They are the unitary operators on Rⁿ.

• The bilateral shift on the sequence space $\backslash ell^2$ indexed by the integers is unitary. In general, any operator in a Hilbert space which acts by shuffling around an orthonormal basis is unitary.

• The Fourier operator is a unitary operator, i.e. the operator which performs the Fourier transform (with proper normalization). This follows from Parseval's theorem.

Properties

• The spectrum of a unitary operator U lies on the unit circle. That is, for any complex number λ in the spectrum, one has |λ|=1. This can be seen as a consequence of the spectral theorem for normal operators. By the theorem, U is unitarily equivalent to multiplication by a Borel-measurable f on L²(μ), for some finite measure space (X, μ). Now U U* = I implies |f(x)|² = 1 μ-a.e. This shows that the essential range of f, therefore the spectrum of U, lies on the unit circle.

Operator theory | Unitary operators