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). Note that the trace operator studied in partial differential equations is an unrelated concept.

Definition

Let H be a separable Hilbert space, an orthonormal basis and A : H \to H a positive bounded linear operator on H. The trace of A is denoted by and defined as independent of the choice of orthonormal basis. A (not necessarily positive) bounded linear operator is called trace class if and only if where denotes the positive-semidefinite Hermitian square root. The trace-norm of a trace class operator T is defined as 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 and this definition of trace of A coincides with the definition 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:

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 such that

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 then the associated Hilbert–Schmidt integral operator T_K is trace class, i.e.,

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 by 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),

Properties

<ol> <li>If A : H \to H is a non-negative [self-adjoint operator](https://bliptext.com/articles/self-adjoint-operator), then A is trace-class [if and only if](https://bliptext.com/articles/if-and-only-if) Therefore, a [self-adjoint operator](https://bliptext.com/articles/self-adjoint-operator) A is trace-class [if and only if](https://bliptext.com/articles/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](https://bliptext.com/articles/continuous-functional-calculus).)</li> <li>The trace is a linear functional over the space of trace-class operators, that is, The bilinear map is an [inner product](https://bliptext.com/articles/inner-product) on the trace class; the corresponding norm is called the [Hilbert–Schmidt](https://bliptext.com/articles/hilbert-schmidt-operator) norm. The completion of the trace-class operators in the Hilbert–Schmidt norm are called the Hilbert–Schmidt operators.</li> <li> is a positive linear functional such that if T is a trace class operator satisfying then T = 0.</li> <li>If T : H \to H is trace-class then so is T^* and </li> <li>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](https://bliptext.com/articles/ideal-ring-theory) 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.</li> <li>If and are two orthonormal bases of H and if T is trace class then </li> <li>If A is trace-class, then one can define the [Fredholm determinant](https://bliptext.com/articles/fredholm-determinant) of I + A: where is the spectrum of A. The trace class condition on A guarantees that the infinite product is finite: indeed, It also implies that if and only if (I + A) is invertible.</li> <li>If A : H \to H is trace class then for any [orthonormal basis](https://bliptext.com/articles/orthonormal-basis) of H, the sum of positive terms is finite.</li> <li>If A = B^* C for some [Hilbert-Schmidt operators](https://bliptext.com/articles/hilbert-schmidt-operator) B and C then for any normal vector e \in H, holds.</li> </ol>

Lidskii's theorem

Let A be a trace-class operator in a separable Hilbert space H, and let be the eigenvalues of A. Let us assume that 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 ). Lidskii's theorem (named after Victor Borisovich Lidskii) states that Note that the series on the right converges absolutely due to Weyl's inequality between the eigenvalues and the singular values 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 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 of non-negative numbers with such that Making the above heuristic comments more precise, we have that T is trace-class iff the series is convergent, T is Hilbert–Schmidt iff is convergent, and T 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 H 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 is By classical inequalities regarding sequences, 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 we identify f with the operator T_f defined by where S_{x,y} is the rank-one operator given by 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 where I is the identity operator: 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 Thus K(H)^ is isometrically isomorphic to B_1.

As the predual of bounded operators

Recall that the dual of \ell^1(\N) is 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 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) the dual space of B_1. This can be used to define the weak-* topology on B(H).