In
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defi ...
, a unitary operator is a
surjective
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element o ...
bounded operator on a
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
that preserves the
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
. Unitary operators are usually taken as operating ''on'' a Hilbert space, but the same notion serves to define the concept of
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
''between'' Hilbert spaces.
A unitary element is a generalization of a unitary operator. In a
unital algebra, an element of the algebra is called a unitary element if ,
where is the identity element.
Definition
Definition 1. A ''unitary operator'' is a
bounded linear operator on a Hilbert space that satisfies , where is the
adjoint of , and is the
identity
Identity may refer to:
* Identity document
* Identity (philosophy)
* Identity (social science)
* Identity (mathematics)
Arts and entertainment Film and television
* ''Identity'' (1987 film), an Iranian film
* ''Identity'' (2003 film), an ...
operator.
The weaker condition defines an ''
isometry
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' ...
''. The other condition, , defines a ''coisometry''. Thus a unitary operator is a bounded linear operator which is both an isometry and a coisometry, or, equivalently, a
surjective
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element o ...
isometry.
An equivalent definition is the following:
Definition 2. A ''unitary operator'' is a bounded linear operator on a Hilbert space for which the following hold:
* is
surjective
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element o ...
, and
* preserves the
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
of the Hilbert space, . In other words, for all
vectors and in 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
The following, seemingly weaker, definition is also equivalent:
Definition 3. A ''unitary operator'' is a bounded linear operator on a Hilbert space for which the following hold:
*the range of is
dense in , and
* preserves the inner product of the Hilbert space, . In other words, for all vectors and in we have:
*:
To see that Definitions 1 & 3 are equivalent, notice that preserving the inner product implies is an
isometry
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' ...
(thus, a
bounded linear operator). The fact that has dense range ensures it has a bounded inverse . It is clear that .
Thus, unitary operators are just
automorphisms of Hilbert spaces, i.e., they preserve the structure (the linear space structure, the inner product, and hence the
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
) of the space on which they act. The
group of all unitary operators from a given Hilbert space to itself is sometimes referred to as the ''Hilbert group'' of , denoted or .
Examples
* The
identity function
Graph of the identity function on the real numbers
In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, un ...
is trivially a unitary operator.
* Rotations in 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 .
* On the
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
of
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s, multiplication by a number of
absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
, that is, a number of the form for , is a unitary operator. is referred to as a phase, and this multiplication is referred to as multiplication by a phase. Notice that the value of modulo does not affect the result of the multiplication, and so the independent unitary operators on are parametrized by a circle. The corresponding group, which, as a set, is the circle, is called .
* More generally,
unitary matrices are precisely the unitary operators on finite-dimensional
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
s, 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. They are the unitary operators on .
* The
bilateral shift on the
sequence space indexed by the
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s is unitary. In general, any operator in a Hilbert space which acts by permuting an
orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For examp ...
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 which performs the
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...
(with proper normalization). This follows from
Parseval's theorem.
* Unitary operators are used in
unitary representations.
*
Quantum logic gate
In quantum computing and specifically the quantum circuit model of computation, a quantum logic gate (or simply quantum gate) is a basic quantum circuit operating on a small number of qubits. They are the building blocks of quantum circuits, ...
s are unitary operators. Not all gates are
Hermitian.
Linearity
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 you obtain
:
Properties
* The
spectrum
A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors ...
of a unitary operator lies on the unit circle. That is, for any complex number in the spectrum, one has . This can be seen as a consequence of the
spectral theorem
In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful be ...
for
normal operators. By the theorem, is unitarily equivalent to multiplication by a Borel-measurable on , for some finite measure space . Now implies , -a.e. This shows that the essential range of , therefore the spectrum of , lies on the unit circle.
* A linear map is unitary if it is surjective and isometric. (Use
Polarization identity
In linear algebra, a branch of mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space.
If a norm arises from an inner product the ...
to show the only if part.)
See also
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*
*
Footnotes
References
*
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{{Hilbert space
Operator theory
Unitary operators