Infunctional analysis, aunitary operator is asurjectivebounded operator on aHilbert space that preserves theinner product.Non-trivial examples include rotations, reflections, and theFourier operator.Unitary operators generalizeunitary matrices.Unitary operators are usually taken as operatingon a Hilbert space, but the same notion serves to define the concept ofisomorphismbetween Hilbert spaces.

Definition 1. Aunitary operator is abounded linear operatorU :H →H on a Hilbert spaceH that satisfiesU*U =UU* =I, whereU* is theadjoint ofU, andI :H →H is theidentity operator.

The weaker conditionU*U =I defines anisometry. The other weaker condition,UU* =I, defines acoisometry. Thus a unitary operator is a bounded linear operator that is both an isometry and a coisometry,^[1] or, equivalently, asurjective isometry.^[2]

An equivalent definition is the following:

Definition 2. Aunitary operator is a bounded linear operatorU :H →H on a Hilbert spaceH for which the following hold:

• U issurjective, and
• U preserves theinner product of the Hilbert space,H. In other words, for allvectorsx andy inH we have:

  ${\displaystyle ⟨Ux,Uy{⟩}_H=⟨x,y{⟩}_H.}$

The notion of isomorphism in thecategory of Hilbert spaces is captured if domain and range are allowed to differ in this definition. Isometries preserveCauchy sequences; hence thecompleteness property of Hilbert spaces is preserved^[3]

The following, seemingly weaker, definition is also equivalent:

Definition 3. Aunitary operator is a bounded linear operatorU :H →H on a Hilbert spaceH for which the following hold:

• the range ofU isdense inH, and
• U preserves the inner product of the Hilbert space,H. In other words, for all vectorsx andy inH we have:

  ${\displaystyle ⟨Ux,Uy{⟩}_H=⟨x,y{⟩}_H.}$

To see that definitions 1 and 3 are equivalent, notice thatU preserving the inner product impliesU is anisometry (thus, abounded linear operator). The fact thatU has dense range ensures it has a bounded inverseU⁻¹. It is clear thatU⁻¹ =U*.

Thus, unitary operators are justautomorphisms of Hilbert spaces, i.e., they preserve the structure (the vector space structure, the inner product, and hence thetopology) of the space on which they act. Thegroup of all unitary operators from a given Hilbert spaceH to itself is sometimes referred to as theHilbert group ofH, denotedHilb(H) orU(H).

• Theidentity function is trivially a unitary operator.
• Rotations inR² are the simplest nontrivial example of unitary operators. Rotations do not change the length of a vector or the angle between two vectors. This example can be expanded toR³. In even higher dimensions, this can be extended to theGivens rotation.
• Reflections, like theHouseholder transformation.
• ${\displaystyle \frac{1}{\sqrt{n}}}$ times aHadamard matrix.
• In general, any operator in a Hilbert space that acts by permuting anorthonormal basis is unitary. In the finite dimensional case, such operators are thepermutation matrices.
• On thevector spaceC ofcomplex numbers, multiplication by a number ofabsolute value1, that is, a number of the forme^iθ forθ ∈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 ofθ modulo2π does not affect the result of the multiplication, and so the independent unitary operators onC are parametrized by a circle. The corresponding group, which, as a set, is the circle, is calledU(1).
• TheFourier operator is a unitary operator, i.e. the operator that performs theFourier transform (with proper normalization). This follows fromParseval's theorem.
• Quantum logic gates are unitary operators. Not all gates areHermitian.
• More generally,unitary matrices are precisely the unitary operators on finite-dimensionalHilbert spaces, so the notion of a unitary operator is a generalization of the notion of a unitary matrix.Orthogonal matrices are the special case of unitary matrices in which all entries are real.^[4] They are the unitary operators onRⁿ.
• Thebilateral shift on thesequence spaceℓ² indexed by theintegers is unitary.
• Theunilateral shift (right shift) is an isometry; its conjugate (left shift) is a coisometry.
• Unitary operators are used inunitary representations.
• Aunitary element is a generalization of a unitary operator. In aunital algebra, an elementU of the algebra is called a unitary element ifU*U =UU* =I, whereI is the multiplicativeidentity element.^[5]
• Any composition of the above.

Thelinearity requirement in the definition of a unitary operator can be dropped without changing the meaning because it can be derived from linearity andpositive-definiteness of thescalar product:

Analogously we obtain

${\displaystyle \Vert U(x+y)-(Ux+Uy)\Vert =0.}$

• Thespectrum of a unitary operatorU lies on theunit circle. That is, for any complex numberλ in the spectrum, one has|λ| = 1. This can be seen as a consequence of thespectral theorem fornormal operators. By the theorem,U is unitarily equivalent to multiplication by aBorel-measurablef onL²(μ), for some finitemeasure space(X,μ). NowUU* =I implies|f(x)|² = 1,μ-a.e. This shows that the essential range off, therefore the spectrum ofU, lies on the unit circle.
• A linear map is unitary if it is surjective and isometric. (UsePolarization identity to show the only if part.)

• Antiunitary – Bijective antilinear map between two complex Hilbert spacesPages displaying short descriptions of redirect targets
• Crinkled arc
• Quantum logic gate – Basic circuit in quantum computing
• Unitary matrix – Complex matrix whose conjugate transpose equals its inverse
• Unitary transformation – Endomorphism preserving the inner product

• Conway, J. B. (1990).A Course in Functional Analysis. Graduate Texts in Mathematics. Vol. 96.Springer Verlag.ISBN 0-387-97245-5.

• Doran, Robert S.; Belfi, Victor A. (1986).Characterizations of C*-Algebras: The Gelfand-Naimark Theorems. New York: Marcel Dekker.ISBN 0-8247-7569-4.

• Halmos, Paul (1982).A Hilbert space problem book. Graduate Texts in Mathematics. Vol. 19 (2nd ed.). Springer Verlag.ISBN 978-0387906850.

• Lang, Serge (1972).Differential manifolds. Reading, Mass.–London–Don Mills, Ont.: Addison-Wesley Publishing Co., Inc.ISBN 978-0387961132.

• Roman, Stephen (2008),Advanced Linear Algebra,Graduate Texts in Mathematics (Third ed.), Springer,ISBN 978-0-387-72828-5

• Operator theory
• Unitary operators
• Linear operators

• Articles with short description
• Short description matches Wikidata
• Pages displaying short descriptions of redirect targets via Module:Annotated link