Definition 1. A unitary operator is a bounded linear operatorU : H → H on a Hilbert space H that satisfies U*U = UU* = I, where U* is the adjoint of U, and I : H → H is the identity operator.
The weaker condition U*U = I defines an isometry. The other condition, UU* = I, defines a coisometry. Thus a unitary operator is a bounded linear operator that is both an isometry and a coisometry,[1] or, equivalently, a surjective isometry.[2]
An equivalent definition is the following:
Definition 2. A unitary operator is a bounded linear operator U : H → H on a Hilbert space H for which the following hold:
U preserves the inner product of the Hilbert space, H. In other words, for all vectorsx and y in H we have:
The notion of isomorphism in the category of Hilbert spaces is captured if domain and range are allowed to differ in this definition. Isometries preserve Cauchy sequences; hence the completeness property of Hilbert spaces is preserved[3]
The following, seemingly weaker, definition is also equivalent:
Definition 3. A unitary operator is a bounded linear operator U : H → H on a Hilbert space H for which the following hold:
U preserves the inner product of the Hilbert space, H. In other words, for all vectors x and y in H we have:
To see that definitions 1 and 3 are equivalent, notice that U preserving 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 automorphisms of Hilbert spaces, i.e., they preserve the structure (the vector space structure, the inner product, and hence the topology) of the space on which they act. The group of all unitary operators from a given Hilbert space H to itself is sometimes referred to as the Hilbert group of H, denoted Hilb(H) or U(H).
Rotations in R2 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 to R3.
On the vector spaceC of complex numbers, multiplication by a number of absolute value1, that is, a number of the form eiθ 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 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 on Rn.
A unitary element is a generalization of a unitary operator. In a unital algebra, an element U of the algebra is called a unitary element if U*U = UU* = I, where I is the multiplicative identity element.[5]
Linearityedit
The linearity requirement in the definition of a unitary operator can be dropped without changing the meaning because it can be derived from linearity and positive-definiteness of the scalar product:
Analogously we obtain
Propertiesedit
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-measurablef on L2(μ), for some finite measure space(X, μ). Now UU* = 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.
A linear map is unitary if it is surjective and isometric. (Use Polarization identity to show the only if part.)
See alsoedit
Antiunitary – Bijective antilinear map between two complex Hilbert spacesPages displaying short descriptions of redirect targets
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For unitarity in physics see Unitarity physics In functional analysis a unitary operator is a surjective bounded operator on a Hilbert space that preserves the inner product Unitary operators are usually taken as operating on a Hilbert space but the same notion serves to define the concept of isomorphism between Hilbert spaces Contents 1 Definition 2 Examples 3 Linearity 4 Properties 5 See also 6 Footnotes 7 ReferencesDefinition editDefinition 1 A unitary operator is a bounded linear operator U H H on a Hilbert space H that satisfies U U UU I where U is the adjoint of U and I H H is the identity operator The weaker condition U U I defines an isometry The other condition UU I defines a coisometry Thus a unitary operator is a bounded linear operator that is both an isometry and a coisometry 1 or equivalently a surjective isometry 2 An equivalent definition is the following Definition 2 A unitary operator is a bounded linear operator U H H on a Hilbert space H for which the following hold U is surjective and U preserves the inner product of the Hilbert space H In other words for all vectors x and y in H we have U x U y H x y H displaystyle langle Ux Uy rangle H langle x y rangle H nbsp The notion of isomorphism in the category of Hilbert spaces is captured if domain and range are allowed to differ in this definition Isometries preserve Cauchy sequences hence the completeness property of Hilbert spaces is preserved 3 The following seemingly weaker definition is also equivalent Definition 3 A unitary operator is a bounded linear operator U H H on a Hilbert space H for which the following hold the range of U is dense in H and U preserves the inner product of the Hilbert space H In other words for all vectors x and y in H we have U x U y H x y H displaystyle langle Ux Uy rangle H langle x y rangle H nbsp To see that definitions 1 and 3 are equivalent notice that U preserving 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 automorphisms of Hilbert spaces i e they preserve the structure the vector space structure the inner product and hence the topology of the space on which they act The group of all unitary operators from a given Hilbert space H to itself is sometimes referred to as the Hilbert group of H denoted Hilb H or U H Examples editThe identity function is trivially a unitary operator Rotations in R2 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 to R3 On the vector space C of complex numbers multiplication by a number of absolute value 1 that is a number of the form ei8 for 8 R is a unitary operator 8 is referred to as a phase and this multiplication is referred to as multiplication by a phase Notice that the value of 8 modulo 2p 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 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 on Rn The bilateral shift on the sequence space ℓ2 indexed by the integers is unitary In general any operator in a Hilbert space that acts by permuting an orthonormal basis is unitary In the finite dimensional case such operators are the permutation matrices The unilateral shift right shift is an isometry its conjugate left shift is a coisometry The Fourier operator is a unitary operator i e the operator that performs the Fourier transform with proper normalization This follows from Parseval s theorem Unitary operators are used in unitary representations Quantum logic gates are unitary operators Not all gates are Hermitian A unitary element is a generalization of a unitary operator In a unital algebra an element U of the algebra is called a unitary element if U U UU I where I is the multiplicative identity element 5 Linearity editThe linearity requirement in the definition of a unitary operator can be dropped without changing the meaning because it can be derived from linearity and positive definiteness of the scalar product l U x U l x 2 l U x U l x l U x U l x l U x 2 U l x 2 U l x l U x l U x U l x l 2 U x 2 U l x 2 l U l x U x l U x U l x l 2 x 2 l x 2 l l x x l x l x 0 displaystyle begin aligned lambda U x U lambda x 2 amp langle lambda U x U lambda x lambda U x U lambda x rangle 5pt amp lambda U x 2 U lambda x 2 langle U lambda x lambda U x rangle langle lambda U x U lambda x rangle 5pt amp lambda 2 U x 2 U lambda x 2 overline lambda langle U lambda x U x rangle lambda langle U x U lambda x rangle 5pt amp lambda 2 x 2 lambda x 2 overline lambda langle lambda x x rangle lambda langle x lambda x rangle 5pt amp 0 end aligned nbsp Analogously we obtain U x y U x U y 0 displaystyle U x y Ux Uy 0 nbsp Properties editThe spectrum of a unitary operator U lies on the unit circle That is for any complex number l in the spectrum one has l 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 m for some finite measure space X m Now UU I implies f x 2 1 m a e This shows that the essential range of f therefore the spectrum of U lies on the unit circle A linear map is unitary if it is surjective and isometric Use Polarization identity to show the only if part See also editAntiunitary 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 productFootnotes edit Halmos 1982 Sect 127 page 69 Conway 1990 Proposition I 5 2 Conway 1990 Definition I 5 1 Roman 2008 p 238 10 Doran amp Belfi 1986 p 55References editConway 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 Retrieved from https en wikipedia org w index php title Unitary operator amp oldid 1221919545, 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