Editing Trace class (section)

=== 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.