HOME

TheInfoList



OR:

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 H 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 : H \to H 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 H 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 A, denoted by \operatorname A, is the sum of the series\operatorname A = \sum_k \left\langle A e_k, e_k \right\rangle,where \left(e_k\right)_ 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 H. 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 T : H \to H on H, 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 , T, , 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 T^* T, that is, , T, := \sqrt 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 H such that , T, \circ , T, = T^* \circ T. The operator T : H \to H is said to be in the trace class if \operatorname (, T, ) < \infty.We denote the space of all trace class linear operators on by B_1(H). (One can show that this is indeed a vector space.) If T is in the trace class, we define the trace of T by\operatorname T = \sum_k \left\langle T e_k, e_k \right\rangle,where \left(e_k\right)_ is an arbitrary orthonormal basis of H. 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 T : H \to H, each of the following statements is equivalent to T being in the trace class: * \operatorname (, T, ) < \infty. * 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, ...
\left(e_k\right)_ of , the sum of positive terms \sum_k \left\langle , T, \, e_k, e_k \right\rangle 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, ...
\left(e_k\right)_ of , the sum of positive terms \sum_k \left\langle , T, \, e_k, e_k \right\rangle 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 \sum_^ s_i < \infty, where s_1, s_2, \ldots are the eigenvalues of , T, (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 \left(x_i\right)_^ and \left(y_i\right)_^ in H and a sequence \left(\lambda_i\right)_^ in \ell^1 such that for all x \in H, T(x) = \sum_^ \lambda_i \left\langle x, x_i \right\rangle y_i. Here, the infinite sum means that the sequence of partial sums \left(\sum_^ \lambda_i \left\langle x, x_i \right\rangle y_i\right)_^ converges to T(x) 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. * \sqrt 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 A^ and B^ of H^ and H^, 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 ...
\mu on A^ \times B^ of total mass \leq 1 such that for all x \in H and y^ \in H^: y^ (T(x)) = \int_ x^(x) \; y^\left(y^\right) \, \mathrm \mu \left(x^, y^\right).


Trace-norm

We define the trace-norm of a trace class operator to be the value \, T\, _1 := \operatorname (, T, ). 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 B_1(H) and that B_1(H), 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 \, T\, _1 = \sup \left\.


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 B_1(H) (when endowed with the \, \cdot \, _1 norm). The composition of two Hilbert-Schmidt operators is a trace class operator. Given any x, y \in H, define the operator x \otimes y : H \to H by (x \otimes y)(z) := \langle z, y \rangle x. Then x \otimes y 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''), \operatorname(A(x \otimes y)) = \langle A x, y \rangle.


Properties

  1. If A : H \to H 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 A is trace-class if and only if \operatorname A < \infty. Therefore, a self-adjoint operator A 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 A^ and negative part A^ 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' ...
    .)
  2. The trace is a linear functional over the space of trace-class operators, that is, \operatorname(aA + bB) = a \operatorname(A) + b \operatorname(B). The bilinear map \langle A, B \rangle = \operatorname(A^* B) 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.
  3. \operatorname : B_1(H) \to \Complex is a positive linear functional such that if T is a trace class operator satisfying T \geq 0 \text\operatorname T = 0, then T = 0.
  4. If T : H \to H is trace-class then so is T^* and \, T\, _1 = \left\, T^*\right\, _1.
  5. If A : H \to H is bounded, and T : H \to H is trace-class, then AT and TA 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 \, A T\, _1 = \operatorname(, A T, ) \leq \, A\, \, T\, _1, \quad \, T A\, _1 = \operatorname(, T A, ) \leq \, A\, \, T\, _1. Furthermore, under the same hypothesis, \operatorname(A T) = \operatorname(T A) and , \operatorname(A T), \leq \, A\, \, T\, . The last assertion also holds under the weaker hypothesis that ''A'' and ''T'' are Hilbert–Schmidt.
  6. If \left(e_k\right)_ and \left(f_k\right)_ are two orthonormal bases of ''H'' and if ''T'' is trace class then \sum_ \left, \left\langle T e_k, f_k \right\rangle \ \leq \, T\, _.
  7. 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 I + A: \det(I + A) := \prod_ + \lambda_n(A) where \_n is the spectrum of A. The trace class condition on A guarantees that the infinite product is finite: indeed, \det(I + A) \leq e^. It also implies that \det(I + A) \neq 0 if and only if (I + A) is invertible.
  8. If A : H \to H 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, ...
    \left(e_k\right)_ of H, the sum of positive terms \sum_k \left, \left\langle A \, e_k, e_k \right\rangle \ is finite.
  9. If A = B^* C for some Hilbert-Schmidt operators B and C then for any normal vector e \in H, , \langle A e, e \rangle, = \frac \left(\, B e\, ^2 + \, C e\, ^2\right) holds.


Lidskii's theorem

Let A be a trace-class operator in a separable Hilbert space H, and let \_^N, N \leq \infty be the eigenvalues of A. Let us assume that \lambda_n(A) are enumerated with algebraic multiplicities taken into account (that is, if the algebraic multiplicity of \lambda is k, then \lambda is repeated k times in the list \lambda_1(A), \lambda_2(A), \dots). 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 \operatorname(A)=\sum_^N \lambda_n(A) 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 ...
\sum_^N \left, \lambda_n(A)\ \leq \sum_^M s_m(A) between the eigenvalues \_^N 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 \_^M of the compact operator A.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 \ell^1(\N). 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 \ell^1 sequence with respect to some choice of a pair of Hilbert bases. In the same vein, the bounded operators are noncommutative versions of \ell^(\N), the compact operators that of c_0 (the sequences convergent to 0), Hilbert–Schmidt operators correspond to \ell^2(\N), 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 T on a Hilbert space takes the following canonical form: there exist orthonormal bases \left(u_i\right)_ and \left(v_i\right)_ and a sequence \left(\alpha_i\right)_ of non-negative numbers with \alpha_i \to 0 such that T x = \sum_ \alpha_i \langle x, v_i\rangle u_i \quad \text x\in H. Making the above heuristic comments more precise, we have that T is trace-class iff the series \sum_i \alpha_i is convergent, T is Hilbert–Schmidt iff \sum_i \alpha_i^2 is convergent, and T is finite-rank iff the sequence \left(\alpha_i\right)_ has only finitely many nonzero terms. This allows to relate these classes of operators. The following inclusions hold and are all proper when H is infinite-dimensional:\ \subseteq \ \subseteq \ \subseteq \. The trace-class operators are given the trace norm \, T\, _1 = \operatorname \left left(T^* T\right)^\right= \sum_i \alpha_i. The norm corresponding to the Hilbert–Schmidt inner product is \, T\, _2 = \left operatorname \left(T^* T\right)\right = \left(\sum_i \alpha_i^2\right)^. 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 \, T \, = \sup_ \left(\alpha_i\right). By classical inequalities regarding sequences, \, T\, \leq \, T\, _2 \leq \, T\, _1 for appropriate T. 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 c_0 is \ell^1(\N). Similarly, we have that the dual of compact operators, denoted by K(H)^*, is the trace-class operators, denoted by B_1. The argument, which we now sketch, is reminiscent of that for the corresponding sequence spaces. Let f \in K(H)^*, we identify f with the operator T_f defined by \langle T_f x, y \rangle = f\left(S_\right), where S_ is the rank-one operator given by S_(h) = \langle h, y \rangle x. This identification works because the finite-rank operators are norm-dense in K(H). In the event that T_f is a positive operator, for any orthonormal basis u_i, one has \sum_i \langle T_f u_i, u_i \rangle = f(I) \leq \, f\, , where I is the identity operator: I = \sum_i \langle \cdot, u_i \rangle u_i. But this means that T_f 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 T_f need not be positive. A limiting argument using finite-rank operators shows that \, T_f\, _1 = \, f\, . Thus K(H)^* is isometrically isomorphic to C_1.


As the predual of bounded operators

Recall that the dual of \ell^1(\N) is \ell^(\N). In the present context, the dual of trace-class operators B_1 is the bounded operators B(H). More precisely, the set B_1 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 B(H). So given any operator T \in B(H), 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 ...
\varphi_T on B_1 by \varphi_T(A) = \operatorname (AT). This correspondence between bounded linear operators and elements \varphi_T 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 B_1 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 B(H) the dual space of C_1. 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 B(H).


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