Unitary operator

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In functional analysis, a branch of mathematics, 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 the following:

1. The range of U is dense, and
2. U preserves the inner product <  ,  > on the Hilbert space, i.e. for all vectors x and y in the Hilbert space,

To see this, notice that U preserves the inner product implies U is an isometry (thus, a bounded linear operator). The fact that U has dense range ensures it has a bounded inverse U^-1. It is clear that U^-1 = U*.

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.

[edit] 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 2π 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 indexed by the integers is unitary. In general, any operator in a Hilbert space which acts by shuffling around an orthonormal basis is unitary. In the finite dimensional case, such operators are the permutation matrices.

• 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.

• Unitary operators are used in unitary representations.

[edit] 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.