In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, specifically
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 space#Definition, inner product, Norm (mathematics)#Defini ...
, a trace-class operator is a linear operator for which a
trace
Trace may refer to:
Arts and entertainment Music
* Trace (Son Volt album), ''Trace'' (Son Volt album), 1995
* Trace (Died Pretty album), ''Trace'' (Died Pretty album), 1993
* Trace (band), a Dutch progressive rock band
* The Trace (album), ''The ...
may be defined, such that the trace is a finite number independent of the choice of basis used to compute the trace. This trace of trace-class operators generalizes the trace of matrices studied in linear algebra. All trace-class operators are
compact operators.
In quantum mechanics,
mixed states are described by
density matrices
In quantum mechanics, a density matrix (or density operator) is a matrix that describes the quantum state of a physical system. It allows for the calculation of the probabilities of the outcomes of any Measurement in quantum mechanics, measurement ...
, which are certain trace class operators.
Trace-class operators are essentially the same as
nuclear operator
In mathematics, nuclear operators are an important class of linear operators introduced by Alexander Grothendieck in his doctoral dissertation. Nuclear operators are intimately tied to the projective tensor product of two topological vector space ...
s, though many authors reserve the term "trace-class operator" for the special case of nuclear operators on
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 and use the term "nuclear operator" in more general
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
s (such as
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
s).
Note that the
trace operator
In mathematics, the trace operator extends the notion of the restriction of a function to the boundary of its domain to "generalized" functions in a Sobolev space. This is particularly important for the study of partial differential equations with ...
studied in partial differential equations is an unrelated concept.
Definition
Suppose
is 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 ...
and
a
bounded linear operator
In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y.
If X and Y are normed vect ...
on
which is
non-negative (I.e., semi—positive-definite) and
self-adjoint
In mathematics, and more specifically in abstract algebra, an element ''x'' of a *-algebra is self-adjoint if x^*=x. A self-adjoint element is also Hermitian, though the reverse doesn't necessarily hold.
A collection ''C'' of elements of a sta ...
. The trace of
, denoted by
is the sum of the series
where
is 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 example, ...
of
. The trace is a sum on non-negative reals and is therefore a non-negative real or infinity. It can be shown that the trace does not depend on the choice of orthonormal basis.
For an arbitrary bounded linear operator
on
we define its
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), an ...
, denoted by
to be the positive
square root
In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or ⋅ ) is . For example, 4 and −4 are square roots of 16, because .
E ...
of
that is,
is the unique bounded
positive operator In mathematics (specifically linear algebra, operator theory, and functional analysis) as well as physics, a linear operator A acting on an inner product space is called positive-semidefinite (or ''non-negative'') if, for every x \in \mathop(A), \la ...
on
such that
The operator
is said to be in the trace class if
We denote the space of all trace class linear operators on by
(One can show that this is indeed a vector space.)
If
is in the trace class, we define the trace of
by
where
is an arbitrary orthonormal basis of
. It can be shown that this is an
absolutely convergent
In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series \textstyle\sum_^\infty a_n is said ...
series of complex numbers whose sum does not depend on the choice of orthonormal basis.
When is finite-dimensional, every operator is trace class and this definition of trace of coincides with the definition of the
trace of a matrix
In linear algebra, the trace of a square matrix , denoted , is defined to be the sum of elements on the main diagonal (from the upper left to the lower right) of . The trace is only defined for a square matrix ().
It can be proved that the trace ...
.
Equivalent formulations
Given a bounded linear operator
, each of the following statements is equivalent to
being in the trace class:
*
* For some
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 example, ...
of , the sum of positive terms
is finite.
* For every
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 example, ...
of , the sum of positive terms
is finite.
* is a
compact operator
In functional analysis, a branch of mathematics, a compact operator is a linear operator T: X \to Y, where X,Y are normed vector spaces, with the property that T maps bounded subsets of X to relatively compact subsets of Y (subsets with compact ...
and
where
are the eigenvalues of
(also known as the
singular values In mathematics, in particular functional analysis, the singular values, or ''s''-numbers of a compact operator T: X \rightarrow Y acting between Hilbert spaces X and Y, are the square roots of the (necessarily non-negative) eigenvalues of the self- ...
of ) with each eigenvalue repeated as often as its multiplicity.
* There exist two
orthogonal
In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''.
By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
sequences
and
in
and a sequence
in
such that for all
Here, the infinite sum means that the sequence of partial sums
converges to
in .
* is a
nuclear operator
In mathematics, nuclear operators are an important class of linear operators introduced by Alexander Grothendieck in his doctoral dissertation. Nuclear operators are intimately tied to the projective tensor product of two topological vector space ...
.
* is equal to the composition of two
Hilbert-Schmidt operators.
*
is a
Hilbert-Schmidt operator.
* is an
integral operator
An integral operator is an operator that involves integration. Special instances are:
* The operator of integration itself, denoted by the integral symbol
* Integral linear operators, which are linear operators induced by bilinear forms invol ...
.
* There exist weakly closed and
equicontinuous
In mathematical analysis, a family of functions is equicontinuous if all the functions are continuous and they have equal variation over a given neighbourhood, in a precise sense described herein.
In particular, the concept applies to countable fa ...
(and
thus weakly compact) subsets
and
of
and
respectively, and some positive
Radon measure
In mathematics (specifically in measure theory), a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space ''X'' that is finite on all compact sets, outer regular on all Borel se ...
on
of total mass
such that for all
and
:
Trace-norm
We define the trace-norm of a trace class operator to be the value
One can show that the trace-norm is a
norm
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envir ...
on the space of all trace class operators
and that
, with the trace-norm, becomes a
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
.
If is trace class then
Examples
Every bounded linear operator that has a finite-dimensional range (i.e. operators of finite-rank) is trace class;
furthermore, the space of all finite-rank operators is a dense subspace of
(when endowed with the
norm).
The composition of two
Hilbert-Schmidt operators is a trace class operator.
Given any
define the operator
by
Then
is a continuous linear operator of rank 1 and is thus trace class;
moreover, for any bounded linear operator ''A'' on ''H'' (and into ''H''),
Properties
- If is a non-negative
self-adjoint operator
In mathematics, a self-adjoint operator on an infinite-dimensional complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle (equivalently, a Hermitian operator in the finite-dimensional case) is a linear map ''A'' (from ''V'' to itse ...
, then is trace-class if and only if Therefore, a self-adjoint operator is trace-class if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bicondi ...
its positive part and negative part are both trace-class. (The positive and negative parts of a self-adjoint operator are obtained by the continuous functional calculus
In mathematics, particularly in operator theory and C*-algebra theory, a continuous functional calculus is a functional calculus which allows the application of a continuous function to normal elements of a C*-algebra.
Theorem
Theorem. Let ''x' ...
.)
- The trace is a linear functional over the space of trace-class operators, that is,
The bilinear map is an
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two ve ...
on the trace class; the corresponding norm is called the Hilbert–Schmidt norm. The completion of the trace-class operators in the Hilbert–Schmidt norm are called the Hilbert–Schmidt operators.
- is a positive linear functional such that if is a trace class operator satisfying then
- If is trace-class then so is and
- If is bounded, and is trace-class, then and are also trace-class (i.e. the space of trace-class operators on ''H'' is an
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considere ...
in the algebra of bounded linear operators on ''H''), and
Furthermore, under the same hypothesis, and
The last assertion also holds under the weaker hypothesis that ''A'' and ''T'' are Hilbert–Schmidt.
- If and are two orthonormal bases of ''H'' and if ''T'' is trace class then
- If ''A'' is trace-class, then one can define the
Fredholm determinant In mathematics, the Fredholm determinant is a complex-valued function which generalizes the determinant of a finite dimensional linear operator. It is defined for bounded operators on a Hilbert space which differ from the identity operator by a ...
of : where is the spectrum of The trace class condition on guarantees that the infinite product is finite: indeed,
It also implies that if and only if is invertible.
- If is trace class then for any
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 example, ...
of the sum of positive terms is finite.
- If for some Hilbert-Schmidt operators and then for any normal vector holds.
Lidskii's theorem
Let
be a trace-class operator in a separable Hilbert space
and let
be the eigenvalues of
Let us assume that
are enumerated with algebraic multiplicities taken into account (that is, if the algebraic multiplicity of
is
then
is repeated
times in the list
). Lidskii's theorem (named after
Victor Borisovich Lidskii Victor Borisovich Lidskii (russian: Виктор Борисович Лидский, 4 May 1924, Odessa – 29 July 2008, Moscow) was a Soviet and Russian mathematician who worked in spectral theory, operator theory, and shell theory. Lidskii disco ...
) states that
Note that the series on the right converges absolutely due to
Weyl's inequality
In linear algebra, Weyl's inequality is a theorem about the changes to eigenvalues of an Hermitian matrix that is perturbed. It can be used to estimate the eigenvalues of a perturbed Hermitian matrix.
Weyl's inequality about perturbation
Let ...
between the eigenvalues
and the
singular value In mathematics, in particular functional analysis, the singular values, or ''s''-numbers of a compact operator T: X \rightarrow Y acting between Hilbert spaces X and Y, are the square roots of the (necessarily non-negative) eigenvalues of the self- ...
s
of the compact operator
[Simon, B. (2005) ''Trace ideals and their applications'', Second Edition, American Mathematical Society.]
Relationship between some classes of operators
One can view certain classes of bounded operators as noncommutative analogue of classical
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 ...
s, with trace-class operators as the noncommutative analogue of the sequence space
Indeed, it is possible to apply the
spectral theorem
In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix (mathematics), matrix can be Diagonalizable matrix, diagonalized (that is, represented as a diagonal matrix i ...
to show that every normal trace-class operator on a separable Hilbert space can be realized in a certain way as an
sequence with respect to some choice of a pair of Hilbert bases. In the same vein, the bounded operators are noncommutative versions of
the
compact operators that of
(the sequences convergent to 0), Hilbert–Schmidt operators correspond to
and
finite-rank operator
In functional analysis, a branch of mathematics, a finite-rank operator is a bounded linear operator between Banach spaces whose range is finite-dimensional.
Finite-rank operators on a Hilbert space A canonical form
Finite-rank operators are ...
s (the sequences that have only finitely many non-zero terms). To some extent, the relationships between these classes of operators are similar to the relationships between their commutative counterparts.
Recall that every compact operator
on a Hilbert space takes the following canonical form: there exist orthonormal bases
and
and a sequence
of non-negative numbers with
such that
Making the above heuristic comments more precise, we have that
is trace-class iff the series
is convergent,
is Hilbert–Schmidt iff
is convergent, and
is finite-rank iff the sequence
has only finitely many nonzero terms. This allows to relate these classes of operators. The following inclusions hold and are all proper when
is infinite-dimensional:
The trace-class operators are given the trace norm
The norm corresponding to the Hilbert–Schmidt inner product is
Also, the usual
operator norm
In mathematics, the operator norm measures the "size" of certain linear operators by assigning each a real number called its . Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces.
Introdu ...
is
By classical inequalities regarding sequences,
for appropriate
It is also clear that finite-rank operators are dense in both trace-class and Hilbert–Schmidt in their respective norms.
Trace class as the dual of compact operators
The
dual space
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by const ...
of
is
Similarly, we have that the dual of compact operators, denoted by
is the trace-class operators, denoted by
The argument, which we now sketch, is reminiscent of that for the corresponding sequence spaces. Let
we identify
with the operator
defined by
where
is the rank-one operator given by
This identification works because the finite-rank operators are norm-dense in
In the event that
is a positive operator, for any orthonormal basis
one has
where
is the identity operator:
But this means that
is trace-class. An appeal to
polar decomposition
In mathematics, the polar decomposition of a square real or complex matrix A is a factorization of the form A = U P, where U is an orthogonal matrix and P is a positive semi-definite symmetric matrix (U is a unitary matrix and P is a positive semi ...
extend this to the general case, where
need not be positive.
A limiting argument using finite-rank operators shows that
Thus
is isometrically isomorphic to
As the predual of bounded operators
Recall that the dual of
is
In the present context, the dual of trace-class operators
is the bounded operators
More precisely, the set
is a two-sided
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considere ...
in
So given any operator
we may define a
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
linear functional
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers).
If is a vector space over a field , the s ...
on
by
This correspondence between bounded linear operators and elements
of the
dual space
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by const ...
of
is an isometric
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 is ...
. It follows that
the dual space of
This can be used to define the
weak-* topology
In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space. The term is most commonly used for the initial topology of a ...
on
See also
*
*
*
Trace operator
In mathematics, the trace operator extends the notion of the restriction of a function to the boundary of its domain to "generalized" functions in a Sobolev space. This is particularly important for the study of partial differential equations with ...
References
Bibliography
*
* Dixmier, J. (1969). ''Les Algebres d'Operateurs dans l'Espace Hilbertien''. Gauthier-Villars.
*
*
{{TopologicalTensorProductsAndNuclearSpaces
Operator theory