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 (for example, Inner product space#Definition, inner product, Norm (mathematics ...
, a unitary operator is a
surjective bounded operator on a
Hilbert space
In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
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, ofte ...
.
Non-trivial examples include rotations, reflections, and the
Fourier operator
The Fourier operator is the integral kernel, kernel of the Fredholm integral equation#Equation of the first kind, Fredholm integral of the first kind that defines the continuous Fourier transform, and is a two-dimensional function when it correspon ...
.
Unitary operators generalize
unitary matrices.
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 or morphism 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 the ...
''between'' Hilbert spaces.
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 operator.
The weaker condition defines an ''
isometry''. The other weaker condition, , defines a ''coisometry''. Thus a unitary operator is a bounded linear operator that is both an isometry and a coisometry, or, equivalently, a
surjective 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, 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, ofte ...
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 and 3 are equivalent, notice that preserving the inner product implies is an
isometry (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
automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphism ...
s of Hilbert spaces, i.e., they preserve the structure (the vector space structure, the inner product, and hence the
topology
Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
) 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 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 . In even higher dimensions, this can be extended to the
Givens rotation.
* Reflections, like the
Householder transformation.
*
times a
Hadamard matrix.
* In general, any operator in a Hilbert space that acts by permuting an
orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space V with finite Dimension (linear algebra), dimension is a Basis (linear algebra), basis for V whose vectors are orthonormal, that is, they are all unit vec ...
is unitary. In the finite dimensional case, such operators are the
permutation matrices
In mathematics, particularly in Matrix (mathematics), matrix theory, a permutation matrix is a square binary matrix that has exactly one entry of 1 in each row and each column with all other entries 0. An permutation matrix can represent a permu ...
.
* On the
vector space
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
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 for ...
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 x 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 .
* The
Fourier operator
The Fourier operator is the integral kernel, kernel of the Fredholm integral equation#Equation of the first kind, Fredholm integral of the first kind that defines the continuous Fourier transform, and is a two-dimensional function when it correspon ...
is a unitary operator, i.e. the operator that performs the
Fourier transform (with proper normalization). This follows from
Parseval's theorem.
*
Quantum logic gates are unitary operators. Not all gates are
Hermitian.
* More generally,
unitary matrices are precisely the unitary operators on finite-dimensional
Hilbert space
In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
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
In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural num ...
indexed by the
integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s is unitary.
* The
unilateral shift (right shift) is an isometry; its conjugate (left shift) is a coisometry.
* Unitary operators are used in
unitary representations.
* 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 multiplicative
identity element
In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is use ...
.
* Any composition of the above.
Linearity
The
linearity
In mathematics, the term ''linear'' is used in two distinct senses for two different properties:
* linearity of a '' function'' (or '' mapping'');
* linearity of a '' polynomial''.
An example of a linear function is the function defined by f(x) ...
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
:
Properties
* The
spectrum
A spectrum (: spectra or spectrums) is a set of related ideas, objects, or properties whose features overlap such that they blend to form a continuum. The word ''spectrum'' was first used scientifically in optics to describe the rainbow of co ...
of a unitary operator lies on the
unit circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
. That is, for any complex number in the spectrum, one has . This can be seen as a consequence of the
spectral theorem
In 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 because computations involvin ...
for
normal operators. By the theorem, is unitarily equivalent to multiplication by a
Borel-measurable on , for some finite
measure space
A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that ...
. 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 to show the only if part.)
See also
*
*
*
*
*
Footnotes
References
*
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{{Hilbert space
Operator theory
Linear operators