Editing Trace class (section)

== 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 {{mvar|H}}.{{sfn|Conway|2000|p=86}}
* {{mvar|T}} is a [[Nuclear_operator#Nuclear_operators_between_Hilbert_spaces|nuclear operator]].{{sfn|Trèves|2006|p=494}}{{sfn|Conway|2000|p=89}}
*: There exist two [[Orthogonal (mathematics)|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 number]]s <math>\left(\lambda_i\right)_{i=1}^{\infty}</math> in [[Sequence_space#ℓp_spaces|<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 value|singular values]] of {{mvar|T}} (or, equivalently, the eigenvalues of <math>|T|</math>), with each value repeated as often as its multiplicity.{{sfn | Reed | Simon | 1980 | pp=203-204,209}}
* {{mvar|T}} is a [[Compact_operator#Compact_operator_on_Hilbert_spaces|compact operator]] with <math>\operatorname{Tr}(|T|)<\infty.</math>
*:If {{mvar|T}} is trace class then{{sfn|Conway|1990|p=268}}
*::<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>
* {{mvar|T}} is an [[Integral linear operator|integral operator]].{{sfn|Trèves|2006|pp=502-508}}
* {{mvar|T}} is equal to the composition of two [[Hilbert-Schmidt operator]]s.{{sfn|Conway|1990|p=267}}
* <math display="inline">\sqrt{|T|}</math> is a Hilbert-Schmidt operator.{{sfn|Conway|1990|p=267}}