Trace class

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Template:Rcatsh 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.Template:Sfn

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 <math>H</math> be a separable Hilbert space, <math>\left\{e_k\right\}_{k=1}^{\infty}</math> an orthonormal basis and <math>A : H \to H</math> a positive bounded linear operator on <math>H</math>. The trace of <math>A</math> is denoted by <math>\operatorname{Tr} (A)</math> and defined asTemplate:SfnTemplate:Sfn

<math>\operatorname{Tr} (A) = \sum_{k=1}^{\infty} \left\langle A e_k, e_k \right\rangle,</math>

independent of the choice of orthonormal basis. A (not necessarily positive) bounded linear operator <math>T:H\rightarrow H</math> is called trace class if and only if

<math>\operatorname{Tr}( |T|) < \infty,</math>

where <math>|T| := \sqrt{T^* T}</math> denotes the positive-semidefinite Hermitian square root.Template:Sfn

The trace-norm of a trace class operator Template:Mvar is defined as <math display="block">\|T\|_1 := \operatorname{Tr} (|T|).</math> One can show that the trace-norm is a norm on the space of all trace class operators <math>B_1(H)</math> and that <math>B_1(H)</math>, with the trace-norm, becomes a Banach space.

When <math>H</math> is finite-dimensional, every (positive) operator is trace class. For <math>A</math> this definition coincides with that of the trace of a matrix. If <math>H</math> is complex, then <math>A</math> is always self-adjoint (i.e. <math>A=A^*=|A|</math>) though the converse is not necessarily true.Template:Sfn

Equivalent formulations

Given a bounded linear operator <math>T : H \to H</math>, each of the following statements is equivalent to <math>T</math> being in the trace class:

• <math display="inline">\operatorname{Tr} (|T|) =\sum_k \left\langle |T| \, e_k, e_k \right\rangle</math> is finite for every orthonormal basis <math>\left(e_k\right)_{k}</math> of Template:Mvar.Template:Sfn
• Template:Mvar is a nuclear operator.Template:SfnTemplate:Sfn

  There exist two orthogonal sequences <math>\left(x_i\right)_{i=1}^{\infty}</math> and <math>\left(y_i\right)_{i=1}^{\infty}</math> in <math>H</math> and positive real numbers <math>\left(\lambda_i\right)_{i=1}^{\infty}</math> in <math>\ell^1</math> such that <math display="inline">\sum_{i=1}^{\infty} \lambda_i < \infty</math> and

  <math>x \mapsto T(x) = \sum_{i=1}^{\infty} \lambda_i \left\langle x, x_i \right\rangle y_i, \quad \forall x \in H,</math>

  where <math>\left(\lambda_i\right)_{i=1}^{\infty}</math> are the singular values of Template:Mvar (or, equivalently, the eigenvalues of <math>|T|</math>), with each value repeated as often as its multiplicity.Template:Sfn

• Template:Mvar is a compact operator with <math>\operatorname{Tr}(|T|)<\infty.</math>

  If Template:Mvar is trace class thenTemplate:Sfn

  <math>\|T\|_1 = \sup \left\{ |\operatorname{Tr} (C T)| : \|C\| \leq 1 \text{ and } C : H \to H \text{ is a compact operator } \right\}.</math>

• Template:Mvar is an integral operator.Template:Sfn
• Template:Mvar is equal to the composition of two Hilbert-Schmidt operators.Template:Sfn
• <math display="inline">\sqrt{|T|}</math> is a Hilbert-Schmidt operator.Template:Sfn

Examples

Spectral theorem

Let <math>T</math> be a bounded self-adjoint operator on a Hilbert space. Then <math>T^2</math> is trace class if and only if <math>T</math> has a pure point spectrum with eigenvalues <math>\left\{\lambda_i(T)\right\}_{i=1}^{\infty}</math> such thatTemplate:Sfn

<math>\operatorname{Tr}(T^2) = \sum_{i=1}^{\infty}\lambda_i(T^2) < \infty.</math>

Mercer's theorem

Mercer's theorem provides another example of a trace class operator. That is, suppose <math>K</math> is a continuous symmetric positive-definite kernel on <math>L^2([a,b])</math>, defined as

<math> K(s,t) = \sum_{j=1}^\infty \lambda_j \, e_j(s) \, e_j(t) </math>

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

<math>\operatorname{Tr}(T_K) = \int_a^b K(t,t)\,dt = \sum_i \lambda_i.</math>

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 <math>B_1(H)</math> (when endowed with the trace norm).Template:Sfn

Given any <math>x, y \in H,</math> define the operator <math> x \otimes y : H \to H</math> by <math>(x \otimes y)(z) := \langle z, y \rangle x.</math> Then <math>x \otimes y</math> 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), <math>\operatorname{Tr}(A(x \otimes y)) = \langle A x, y \rangle.</math>Template:Sfn

Properties

1. If <math>A : H \to H</math> is a non-negative self-adjoint operator, then <math>A</math> is trace-class if and only if <math>\operatorname{Tr} A < \infty.</math> Therefore, a self-adjoint operator <math>A</math> is trace-class if and only if its positive part <math>A^{+}</math> and negative part <math>A^{-}</math> are both trace-class. (The positive and negative parts of a self-adjoint operator are obtained by the continuous functional calculus.)
2. The trace is a linear functional over the space of trace-class operators, that is, <math display="block">\operatorname{Tr}(aA + bB) = a \operatorname{Tr}(A) + b \operatorname{Tr}(B).</math> The bilinear map <math display="block">\langle A, B \rangle = \operatorname{Tr}(A^* B)</math> 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.
3. <math>\operatorname{Tr} : B_1(H) \to \Complex</math> is a positive linear functional such that if <math>T</math> is a trace class operator satisfying <math>T \geq 0 \text{ and }\operatorname{Tr} T = 0,</math> then <math>T = 0.</math>Template:Sfn
4. If <math>T : H \to H</math> is trace-class then so is <math>T^*</math> and <math>\|T\|_1 = \left\|T^*\right\|_1.</math>Template:Sfn
5. If <math>A : H \to H</math> is bounded, and <math>T : H \to H</math> is trace-class, then <math>AT</math> and <math>TA</math> 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), andTemplate:SfnTemplate:Sfn <math display="block">\|A T\|_1 = \operatorname{Tr}(|A T|) \leq \|A\| \|T\|_1, \quad \|T A\|_1 = \operatorname{Tr}(|T A|) \leq \|A\| \|T\|_1.</math> Furthermore, under the same hypothesis,Template:Sfn <math display="block">\operatorname{Tr}(A T) = \operatorname{Tr}(T A)</math> and <math>|\operatorname{Tr}(A T)| \leq \|A\| \|T\|.</math> The last assertion also holds under the weaker hypothesis that A and T are Hilbert–Schmidt.
6. If <math>\left(e_k\right)_{k}</math> and <math>\left(f_k\right)_{k}</math> are two orthonormal bases of H and if T is trace class then <math display="inline">\sum_{k} \left| \left\langle T e_k, f_k \right\rangle \right| \leq \|T\|_{1}.</math>Template:Sfn
7. If A is trace-class, then one can define the Fredholm determinant of <math>I + A</math>: <math display="block">\det(I + A) := \prod_{n \geq 1}[1 + \lambda_n(A)],</math> where <math>\{\lambda_n(A)\}_n</math> is the spectrum of <math>A.</math> The trace class condition on <math>A</math> guarantees that the infinite product is finite: indeed, <math display="block">\det(I + A) \leq e^{\|A\|_1}.</math> It also implies that <math>\det(I + A) \neq 0</math> if and only if <math>(I + A)</math> is invertible.
8. If <math>A : H \to H</math> is trace class then for any orthonormal basis <math>\left(e_k\right)_{k}</math> of <math>H,</math> the sum of positive terms <math display="inline">\sum_k \left| \left\langle A \, e_k, e_k \right\rangle \right|</math> is finite.Template:Sfn
9. If <math>A = B^* C</math> for some Hilbert-Schmidt operators <math>B</math> and <math>C</math> then for any normal vector <math>e \in H,</math> <math display="inline">|\langle A e, e \rangle| = \frac{1}{2} \left(\|B e\|^2 + \|C e\|^2\right)</math> holds.Template:Sfn

Lidskii's theorem

Let <math>A</math> be a trace-class operator in a separable Hilbert space <math>H,</math> and let <math>\{\lambda_n(A)\}_{n=1}^{N\leq \infty}</math> be the eigenvalues of <math>A.</math> Let us assume that <math>\lambda_n(A)</math> are enumerated with algebraic multiplicities taken into account (that is, if the algebraic multiplicity of <math>\lambda</math> is <math>k,</math> then <math>\lambda</math> is repeated <math>k</math> times in the list <math>\lambda_1(A), \lambda_2(A), \dots</math>). Lidskii's theorem (named after Victor Borisovich Lidskii) states that <math display="block">\operatorname{Tr}(A)=\sum_{n=1}^N \lambda_n(A)</math>

Note that the series on the right converges absolutely due to Weyl's inequality <math display="block">\sum_{n=1}^N \left|\lambda_n(A)\right| \leq \sum_{m=1}^M s_m(A)</math> between the eigenvalues <math>\{\lambda_n(A)\}_{n=1}^N</math> and the singular values <math>\{s_m(A)\}_{m=1}^M</math> of the compact operator <math>A.</math><ref>Simon, B. (2005) Trace ideals and their applications, Second Edition, American Mathematical Society.</ref>

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 <math>\ell^1(\N).</math>

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 <math>\ell^1</math> sequence with respect to some choice of a pair of Hilbert bases. In the same vein, the bounded operators are noncommutative versions of <math>\ell^{\infty}(\N),</math> the compact operators that of <math>c_0</math> (the sequences convergent to 0), Hilbert–Schmidt operators correspond to <math>\ell^2(\N),</math> and finite-rank operators to <math>c_{00}</math> (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 <math>T</math> on a Hilbert space takes the following canonical form: there exist orthonormal bases <math>(u_i)_i </math> and <math>(v_i)_i</math> and a sequence <math>\left(\alpha_i\right)_{i}</math> of non-negative numbers with <math>\alpha_i \to 0</math> such that <math display="block">T x = \sum_i \alpha_i \langle x, v_i\rangle u_i \quad \text{ for all } x\in H.</math> Making the above heuristic comments more precise, we have that <math>T</math> is trace-class iff the series <math display="inline">\sum_i \alpha_i</math> is convergent, <math>T</math> is Hilbert–Schmidt iff <math display="inline">\sum_i \alpha_i^2</math> is convergent, and <math>T</math> is finite-rank iff the sequence <math>\left(\alpha_i\right)_{i}</math> has only finitely many nonzero terms. This allows to relate these classes of operators. The following inclusions hold and are all proper when <math>H</math> is infinite-dimensional:<math display="block">\{ \text{ finite rank } \} \subseteq \{ \text{ trace class } \} \subseteq \{ \text{ Hilbert--Schmidt } \} \subseteq \{ \text{ compact } \}.</math>

The trace-class operators are given the trace norm <math display="inline">\|T\|_1 = \operatorname{Tr} \left[\left(T^* T\right)^{1/2}\right] = \sum_i \alpha_i.</math> The norm corresponding to the Hilbert–Schmidt inner product is <math display="block">\|T\|_2 = \left[\operatorname{Tr} \left(T^* T\right)\right]^{1/2} = \left(\sum_i \alpha_i^2\right)^{1/2}.</math> Also, the usual operator norm is <math display="inline">\| T \| = \sup_{i} \left(\alpha_i\right).</math> By classical inequalities regarding sequences, <math display="block">\|T\| \leq \|T\|_2 \leq \|T\|_1</math> for appropriate <math>T.</math>

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 <math>c_0</math> is <math>\ell^1(\N).</math> Similarly, we have that the dual of compact operators, denoted by <math>K(H)^*,</math> is the trace-class operators, denoted by <math>B_1.</math> The argument, which we now sketch, is reminiscent of that for the corresponding sequence spaces. Let <math>f \in K(H)^*,</math> we identify <math>f</math> with the operator <math>T_f</math> defined by <math display="block">\langle T_f x, y \rangle = f\left(S_{x,y}\right),</math> where <math>S_{x,y}</math> is the rank-one operator given by <math display="block">S_{x,y}(h) = \langle h, y \rangle x.</math>

This identification works because the finite-rank operators are norm-dense in <math>K(H).</math> In the event that <math>T_f</math> is a positive operator, for any orthonormal basis <math>u_i,</math> one has <math display="block">\sum_i \langle T_f u_i, u_i \rangle = f(I) \leq \|f\|,</math> where <math>I</math> is the identity operator: <math display="block">I = \sum_i \langle \cdot, u_i \rangle u_i.</math>

But this means that <math>T_f</math> is trace-class. An appeal to polar decomposition extend this to the general case, where <math>T_f</math> need not be positive.

A limiting argument using finite-rank operators shows that <math>\|T_f\|_1 = \|f\|.</math> Thus <math>K(H)^*</math> is isometrically isomorphic to <math>B_1.</math>

As the predual of bounded operators

Recall that the dual of <math>\ell^1(\N)</math> is <math>\ell^{\infty}(\N).</math> In the present context, the dual of trace-class operators <math>B_1</math> is the bounded operators <math>B(H).</math> More precisely, the set <math>B_1</math> is a two-sided ideal in <math>B(H).</math> So given any operator <math>T \in B(H),</math> we may define a continuous linear functional <math>\varphi_T</math> on <math>B_1</math> by <math>\varphi_T(A) = \operatorname{Tr} (AT).</math> This correspondence between bounded linear operators and elements <math>\varphi_T</math> of the dual space of <math>B_1</math> is an isometric isomorphism. It follows that <math>B(H)</math> Template:Em the dual space of <math>B_1.</math> This can be used to define the weak-* topology on <math>B(H).</math>

See also

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Bibliography

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• Dixmier, J. (1969). Les Algebres d'Operateurs dans l'Espace Hilbertien. Gauthier-Villars.
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• Template:Trèves François Topological vector spaces, distributions and kernels