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| In [[mathematics]], a '''trace class''' operator is a [[compact operator]] for which a [[Trace (linear algebra)|trace]] may be defined, such that the trace is finite and independent of the choice of basis.
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| Trace class operators are essentially the same as [[nuclear operator]]s, though many authors reserve the term "trace class operator" for the special case of nuclear operators on [[Hilbert space]]s, and reserve nuclear (=trace class) operators for more general [[Banach space]]s.
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| ==Definition==
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| Mimicking the definition for matrices, a [[bounded linear operator]] ''A'' over a [[separable space|separable]] [[Hilbert space]] ''H'' is said to be in the '''trace class''' if for some (and hence all) [[orthonormal basis|orthonormal bases]] {''e''<sub>''k''</sub>}<sub>''k''</sub> of ''H'' the sum of positive terms
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| :<math>\|A\|_{1}= {\rm Tr}|A|:=\sum_{k} \langle (A^*A)^{1/2} \, e_k, e_k \rangle </math> | |
| is finite.
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| In this case, the sum
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| :<math>{\rm Tr} A:=\sum_{k} \langle A e_k, e_k \rangle</math>
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| is [[absolutely convergent]] and is independent of the choice of the orthonormal basis. This value is called the '''trace''' of ''A''. When ''H'' is finite-dimensional, every operator is trace class and this definition of trace of ''A'' coincides with the definition of the [[trace (matrix)| trace of a matrix]].
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| By extension, if ''A'' is a non-negative self-adjoint operator, we can also define the trace of ''A'' as an extended real number by the
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| possibly divergent sum
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| :<math>\sum_{k} \langle A e_k, e_k \rangle. </math>
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| ==Properties== | |
| {| cellspacing="0" cellpadding="2"
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| |valign="top"| '''1.''' || If ''A'' is a non-negative self-adjoint, ''A'' is trace class if and only if Tr(''A'') < ∞. Therefore a self adjoint operator ''A'' is trace class [[if and only if]] its positive part ''A''<sup>+</sup> and negative part ''A''<sup>−</sup> are both trace class. (The positive and negative parts of a self adjoint operator are obtained via the [[continuous functional calculus]].)
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| |valign="top"| '''2.''' || The trace is a linear functional over the space of trace class operators, i.e.
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| :<math>\operatorname{Tr}(aA+bB)=a\,\operatorname{Tr}(A)+b\,\operatorname{Tr}(B).</math>
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| The bilinear map
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| :<math> \langle A, B \rangle = \operatorname{Tr}(A^* B) </math>
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| 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.
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| |valign="top"| '''3.''' || If <math>A</math> is bounded and <math>B</math> is trace class, <math>AB</math> and <math>BA</math> are also trace class and <ref>M. Reed and B. Simon Functional Analysis, Exercises 27, 28 page 218</ref>
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| :<math> \|AB\|_1=\operatorname{Tr}(|AB|)\le \|A\|\|B\|_1,\qquad \|BA\|_1=\operatorname{Tr}(|BA|)\le \|A\|\|B\|_1</math>
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| besides, under the same hypothesis,
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| :<math> \operatorname{Tr}(AB)=\operatorname{Tr}(BA)</math>
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| The last assertion also holds under the weaker hypothesis that <math>A</math> and <math>B</math> are Hilbert Schmidt.
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| |valign="top"| '''4.''' || If <math>A</math> is trace class, then one can define the [[Fredholm determinant]] of <math>1+A</math>
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| :<math> {\rm det} (I+A):=\prod_{n\ge 1}[1+\lambda_n(A)]</math>
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| for <math>\{\lambda_n(A)\}_n</math> the elements of the spectrum of <math>A</math>; the trace class condition on <math>A</math> guarantees that the infinite product is finite: indeed
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| :<math> {\rm det} (I+A)\le e^{\|A\|_1};</math>
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| it also guarantees that <math> {\rm det} (I+A)\neq 0</math> if and only if <math> (I+A)</math> is invertible. | |
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| |}
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| === Lidskii's theorem ===
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| Let <math>A</math> be a trace class operator in a separable Hilbert space <math>H</math>, and let
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| <math>\{\lambda_n(A)\}_{n=1}^N,</math> <math>N\leq \infty</math>
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| be the eigenvalues of <math>A</math>.
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| Let us assume that <math>\lambda_n(A)</math>
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| are enumerated with algebraic multiplicities taken into account
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| (i.e. if the algebraic multiplicity of <math>\lambda</math>
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| is <math>k</math> then <math>\lambda</math> is
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| repeated <math>k</math> times in the list
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| <math>\lambda_1(A),\lambda_2(A),\dots</math>).
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| Lidskii's theorem (named after [[Victor Borisovich Lidskii]]) states that
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| :<math>\sum_{n=1}^N \lambda_n(A)=\operatorname{Tr}(A).</math> | |
| Note that the series in the left hand side converges absolutely
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| due to Weyl's inequality
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| :<math>\sum_{n=1}^N |\lambda_n(A)|\leq \sum_{m=1}^M s_m(A)</math>
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| between the eigenvalues
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| <math>\{\lambda_n(A)\}_{n=1}^N</math> and the
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| [[singular value]]s
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| <math>\{s_m(A)\}_{m=1}^M</math>
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| of a compact operator <math>A</math>.
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| See e.g. <ref>Simon, B. (2005) ''Trace ideals and their applications'', Second Edition, Amer. Math. Soc.</ref>
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| === Relationship between some classes of operators === | |
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| 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 ''l''<sup>1</sup>('''N'''). Indeed, applying the [[spectral theorem]], every normal trace-class operator on a separable Hilbert space can be realized as a ''l''<sup>1</sup> sequence. In the same vein, the bounded operators are noncommutative versions of ''l''<sup>∞</sup>('''N'''), the [[Compact operator on Hilbert space|compact operator]]s that of ''c''<sub>0</sub> (the sequences convergent to 0), Hilbert-Schmidt operators correspond to ''l''<sup>2</sup>('''N'''), and [[finite-rank operator]]s 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.
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| Recall that every compact operator ''T'' on a Hilbert space takes the following canonical form
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| :<math>\forall h \in H, \; T h = \sum _{i = 1} \alpha_i \langle h, v_i\rangle u_i \quad \mbox{where} \quad \alpha_i \geq 0 \quad \mbox{and} \quad \alpha_i \rightarrow 0 </math>
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| for some orthonormal bases {''u<sub>i</sub>''} and {''v<sub>i</sub>''}. Making the above heuristic comments more precise, we have that ''T'' is trace class if the series ∑<sub>''i''</sub> ''α<sub>i</sub>'' is convergent, ''T'' is Hilbert-Schmidt if ∑<sub>''i''</sub> ''α<sub>i</sub>''<sup>2</sup> is convergent, and ''T'' is finite rank if the sequence | |
| {''α<sub>i</sub>''} has only finitely many nonzero terms.
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| The above description allows one to obtain easily some facts that relate these classes of operators. For example, the following inclusions hold and they are all proper when ''H'' is infinite dimensional: {finite rank} ⊂ {trace class} ⊂ {Hilbert-Schmidt} ⊂ {compact}.
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| The trace-class operators are given the trace norm ||''T''||<sub>1</sub> = Tr [ (''T*T'')<sup>½</sup> ] = ∑<sub>''i''</sub> ''α<sub>i</sub>''. The norm corresponding to the Hilbert-Schmidt inner product is ||''T''||<sub>2</sub> = (Tr ''T*T'')<sup>½</sup> = (∑<sub>''i''</sub>''α<sub>i</sub>''<sup>2</sup>)<sup>½</sup>. Also, the usual [[operator norm]] is ||''T''|| = sup<sub>''i''</sub>(''α<sub>i</sub>''). By classical inequalities regarding sequences,
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| :<math>\|T\| \leq \|T\|_2 \leq \|T\|_1 ,</math>
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| for appropriate ''T''.
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| It is also clear that finite-rank operators are dense in both trace-class and Hilbert-Schmidt in their respective norms.
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| === Trace class as the dual of compact operators ===
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| The dual space of ''c''<sub>0</sub> is ''l''<sup>1</sup>('''N'''). Similarly, we have that the dual of compact operators, denoted by ''K''(''H'')*, is the trace-class operators, denoted by ''C''<sub>1</sub>. The argument, which we now sketch, is reminiscent of that for the corresponding sequence spaces. Let ''f'' ∈ ''K''(''H'')*, we identify ''f'' with the operator ''T<sub>f</sub>'' defined by
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| :<math>\langle T_f x, y \rangle = f(S_{x,y}),</math> | |
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| where ''S''<sub>''x,y''</sub> is the rank-one operator given by
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| :<math>S_{x,y}(h) = \langle h, y \rangle x.</math>
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| This identification works because the finite-rank operators are norm-dense in ''K''(''H''). In the event that ''T<sub>f</sub>'' is a positive operator, for any orthonormal basis ''u<sub>i</sub>'', one has
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| :<math>\sum_i \langle T_f u_i, u_i \rangle = f(I) \leq \|f\|,</math>
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| where ''I'' is the identity operator
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| :<math>I = \sum_i \langle \cdot, u_i \rangle u_i .</math>
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| But this means ''T<sub>f</sub>'' is trace-class. An appeal to [[polar decomposition]] extend this to the general case where ''T<sub>f</sub>'' need not be positive.
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| A limiting argument via finite-rank operators shows that ||''T<sub>f</sub>'' ||<sub>1</sub> = || ''f'' ||. Thus ''K''(''H'')* is isometrically isomorphic to ''C''<sub>1</sub>.
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| === As the predual of bounded operators ===
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| Recall that the dual of ''l''<sup>1</sup>('''N''') is ''l''<sup>∞</sup>('''N'''). In the present context, the dual of trace-class operators ''C''<sub>1</sub> is the bounded operators B(''H''). More precisely, the set ''C''<sub>1</sub> is a two-sided [[ideal (ring theory)|ideal]] in B(''H''). So given any operator ''T'' in B(''H''), we may define a [[continuous function (topology)|continuous]] [[linear functional]] φ<sub>''T''</sub> on <math>C_1</math> by φ<sub>''T''</sub>(''A'')=Tr(''AT''). This correspondence between elements φ<sub>''T''</sub> of the [[dual space]] of <math>C_1</math> and bounded linear operators is an isometric [[isomorphism]]. It follows that B(''H'') ''is'' the dual space of <math>C_1</math>. This can be used to define the [[weak-star operator topology|weak-* topology]] on B(''H'').
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| == Notes ==
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| <references />
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| == References ==
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| #Dixmier, J. (1969). ''Les Algebres d'Operateurs dans l'Espace Hilbertien''. Gauthier-Villars.
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| {{Functional Analysis}}
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| [[Category:Operator theory]]
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