Trace class
Compact operator for which a finite trace can be defined
title: "Trace class" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["operator-theory", "topological-tensor-products", "linear-operators"] description: "Compact operator for which a finite trace can be defined" topic_path: "philosophy" source: "https://en.wikipedia.org/wiki/Trace_class" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0
::summary Compact operator for which a finite trace can be defined ::
In mathematics, specifically functional analysis, a trace-class operator is a linear operator for which a trace 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, quantum states are described by density matrices, which are certain trace class operators.
Trace-class operators are essentially the same as nuclear operators, though many authors reserve the term "trace-class operator" for the special case of nuclear operators on Hilbert spaces and use the term "nuclear operator" in more general topological vector spaces (such as Banach spaces).
Definition
Let H be a separable Hilbert space, \left{e_k\right}{k=1}^{\infty} an orthonormal basis and A : H \to H a positive bounded linear operator on H. The trace of A is denoted by \operatorname{Tr} (A) and defined as :\operatorname{Tr} (A) = \sum{k=1}^{\infty} \left\langle A e_k, e_k \right\rangle, independent of the choice of orthonormal basis. A (not necessarily positive) bounded linear operator T:H\rightarrow H is called trace class if and only if :\operatorname{Tr}( |T|) where |T| := \sqrt{T^* T} denotes the positive-semidefinite Hermitian square root.
The trace-norm of a trace class operator T is defined as |T|_1 := \operatorname{Tr} (|T|). One can show that the trace-norm is a norm on the space of all trace class operators B_1(H) and that B_1(H), with the trace-norm, becomes a Banach space.
When H is finite-dimensional, every (positive) operator is trace class. For A this definition coincides with that of the trace of a matrix. If H is complex, then A is always self-adjoint (i.e. A=A^*=|A|) though the converse is not necessarily true.
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{Tr} (|T|) =\sum_k \left\langle |T| , e_k, e_k \right\rangle is finite for every orthonormal basis \left(e_k\right)_{k} of H.
- T is a nuclear operator.
- : There exist two orthogonal sequences \left(x_i\right){i=1}^{\infty} and \left(y_i\right){i=1}^{\infty} in H and positive real numbers \left(\lambda_i\right){i=1}^{\infty} in \ell^1 such that \sum{i=1}^{\infty} \lambda_i and
- ::x \mapsto T(x) = \sum_{i=1}^{\infty} \lambda_i \left\langle x, x_i \right\rangle y_i, \quad \forall x \in H,
- :where \left(\lambda_i\right)_{i=1}^{\infty} are the singular values of T (or, equivalently, the eigenvalues of |T|), with each value repeated as often as its multiplicity.
- T is a compact operator with \operatorname{Tr}(|T|)
- :If T is trace class then
- ::|T|_1 = \sup \left{ |\operatorname{Tr} (C T)| : |C| \leq 1 \text{ and } C : H \to H \text{ is a compact operator } \right}.
- T is an integral operator.
- T is equal to the composition of two Hilbert-Schmidt operators.
- \sqrt{|T|} is a Hilbert-Schmidt operator.
Examples
Spectral theorem
Let T be a bounded self-adjoint operator on a Hilbert space. Then T^2 is trace class if and only if T has a pure point spectrum with eigenvalues \left{\lambda_i(T)\right}{i=1}^{\infty} such that :\operatorname{Tr}(T^2) = \sum{i=1}^{\infty}\lambda_i(T^2)
Mercer's theorem
Mercer's theorem provides another example of a trace class operator. That is, suppose K is a continuous symmetric positive-definite kernel on L^2([a,b]), defined as : K(s,t) = \sum_{j=1}^\infty \lambda_j , e_j(s) , e_j(t) then the associated Hilbert–Schmidt integral operator T_K is trace class, i.e., :\operatorname{Tr}(T_K) = \int_a^b K(t,t),dt = \sum_i \lambda_i.
Finite-rank operators
Every finite-rank operator is a trace-class operator. Furthermore, the space of all finite-rank operators is a dense subspace of B_1(H) (when endowed with the trace norm).
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{Tr}(A(x \otimes y)) = \langle A x, y \rangle.
Properties
If A : H \to H is a non-negative self-adjoint operator, then A is trace-class if and only if \operatorname{Tr} A Therefore, a self-adjoint operator A is trace-class if and only if 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.)
The trace is a linear functional over the space of trace-class operators, that is, \operatorname{Tr}(aA + bB) = a \operatorname{Tr}(A) + b \operatorname{Tr}(B). The bilinear map \langle A, B \rangle = \operatorname{Tr}(A^* B) is an inner product 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.
\operatorname{Tr} : B_1(H) \to \Complex is a positive linear functional such that if T is a trace class operator satisfying T \geq 0 \text{ and }\operatorname{Tr} T = 0, then T = 0.
If T : H \to H is trace-class then so is T^* and |T|_1 = \left|T^*\right|_1.
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 a two-sided ideal in the algebra of bounded linear operators on H), and |A T|_1 = \operatorname{Tr}(|A T|) \leq |A| |T|_1, \quad |T A|_1 = \operatorname{Tr}(|T A|) \leq |A| |T|_1. Furthermore, under the same hypothesis, \operatorname{Tr}(A T) = \operatorname{Tr}(T A) and |\operatorname{Tr}(A T)| \leq |A| |T|. The last assertion also holds under the weaker hypothesis that A and T are Hilbert–Schmidt.
If \left(e_k\right){k} and \left(f_k\right){k} are two orthonormal bases of H and if T is trace class then \sum_{k} \left| \left\langle T e_k, f_k \right\rangle \right| \leq |T|_{1}.
If A is trace-class, then one can define the Fredholm determinant of I + A: \det(I + A) := \prod_{n \geq 1}[1 + \lambda_n(A)], where {\lambda_n(A)}_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^{|A|_1}. It also implies that \det(I + A) \neq 0 if and only if (I + A) is invertible.
If A : H \to H is trace class then for any orthonormal basis \left(e_k\right)_{k} of H, the sum of positive terms \sum_k \left| \left\langle A , e_k, e_k \right\rangle \right| is finite.
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{1}{2} \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 {\lambda_n(A)}{n=1}^{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) states that \operatorname{Tr}(A)=\sum{n=1}^N \lambda_n(A)
Note that the series on the right converges absolutely due to Weyl's inequality \sum_{n=1}^N \left|\lambda_n(A)\right| \leq \sum_{m=1}^M s_m(A) between the eigenvalues {\lambda_n(A)}{n=1}^N and the singular values {s_m(A)}{m=1}^M of the compact operator A.
Relationship between common classes of operators
One can view certain classes of bounded operators as noncommutative analogue of classical sequence spaces, with trace-class operators as the noncommutative analogue of the sequence space \ell^1(\N).
Indeed, it is possible to apply the spectral theorem 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^{\infty}(\N), the compact operators that of c_0 (the sequences convergent to 0), Hilbert–Schmidt operators correspond to \ell^2(\N), and finite-rank operators to c_{00} (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 (u_i)i and (v_i)i and a sequence \left(\alpha_i\right){i} of non-negative numbers with \alpha_i \to 0 such that T x = \sum_i \alpha_i \langle x, v_i\rangle u_i \quad \text{ for all } 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){i} 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:{ \text{ finite rank } } \subseteq { \text{ trace class } } \subseteq { \text{ Hilbert--Schmidt } } \subseteq { \text{ compact } }.
The trace-class operators are given the trace norm |T|_1 = \operatorname{Tr} \left[\left(T^* T\right)^{1/2}\right] = \sum_i \alpha_i. The norm corresponding to the Hilbert–Schmidt inner product is |T|2 = \left[\operatorname{Tr} \left(T^* T\right)\right]^{1/2} = \left(\sum_i \alpha_i^2\right)^{1/2}. Also, the usual operator norm is | T | = \sup{i} \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 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_{x,y}\right), where S_{x,y} is the rank-one operator given by S_{x,y}(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 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 B_1.
As the predual of bounded operators
Recall that the dual of \ell^1(\N) is \ell^{\infty}(\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 in B(H). So given any operator T \in B(H), we may define a continuous linear functional \varphi_T on B_1 by \varphi_T(A) = \operatorname{Tr} (AT). This correspondence between bounded linear operators and elements \varphi_T of the dual space of B_1 is an isometric isomorphism. It follows that B(H) is the dual space of B_1. This can be used to define the weak-* topology on B(H).
References
Bibliography
- Dixmier, J. (1969). Les Algebres d'Operateurs dans l'Espace Hilbertien. Gauthier-Villars.
References
- Simon, B. (2005) ''Trace ideals and their applications'', Second Edition, American Mathematical Society.
::callout[type=info title="Wikipedia Source"] This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page. ::