Editing Trace class (section)

== Definition ==
Let <math>H</math> be a [[Separable space|separable]] [[Hilbert space]], <math>\left\{e_k\right\}_{k=1}^{\infty}</math> an [[orthonormal basis]] and <math>A : H \to H</math> a [[Positive operator (Hilbert space)|positive]] [[bounded linear operator]] on <math>H</math>. The '''trace''' of <math>A</math> is denoted by <math>\operatorname{Tr} (A)</math> and defined as{{sfn | Conway | 2000 | p=86}}{{sfn | Reed | Simon | 1980 | p=206}}
:<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 [[Conjugate_transpose|Hermitian]] [[square root of a matrix|square root]].{{sfn | Reed | Simon | 1980 | p=196}}

The '''trace-norm''' of a trace class operator {{mvar|T}} is defined as
<math display="block">\|T\|_1 := \operatorname{Tr} (|T|).</math>
One can show that the trace-norm is a [[Norm (mathematics)|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 (matrix)|trace of a matrix]]. If <math>H</math> is complex, then <math>A</math> is always [[Self-adjoint_operator#Bounded_self-adjoint_operators|self-adjoint]] (i.e. <math>A=A^*=|A|</math>) though the converse is not necessarily true.{{sfn | Reed | Simon | 1980 | p=195}}