Editing Trace class (section)

== Properties ==

<ol>
<li>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]].)</li>

<li>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 operator|Hilbert–Schmidt]] norm. The completion of the trace-class operators in the Hilbert–Schmidt norm are called the Hilbert–Schmidt operators.</li>

<li><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>{{sfn|Conway|1990|p=267}}</li>

<li>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>{{sfn|Conway|1990|p=267}}</li>

<li>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 (ring theory)|ideal]] in the algebra of bounded linear operators on ''H''), and{{sfn|Conway|1990|p=267}}{{sfn | Reed | Simon | 1980 | p=218}}
<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,{{sfn|Conway|1990|p=267}} <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.</li>

<li>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>{{sfn|Conway|1990|p=268}}</li>

<li>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.</li>

<li>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.{{sfn|Conway|1990|p=267}}</li>

<li>If <math>A = B^* C</math> for some [[Hilbert-Schmidt operator]]s <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.{{sfn|Conway|1990|p=267}}</li>
</ol>

=== 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 value]]s <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 space]]s, 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 operator on Hilbert space|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 operator]]s 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 (ring theory)|ideal]] in <math>B(H).</math> So given any operator <math>T \in B(H),</math> we may define a [[Continuous linear operator|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> {{em|is}} the dual space of <math>B_1.</math> This can be used to define the [[Weak-star operator topology|weak-* topology]] on <math>B(H).</math>