Trace class

In mathematics, a trace class operator is a compact operator for which a trace may be defined, such that the trace is finite and independent of the choice of basis. Trace class operators are essentially the same as nuclear operators, though many authors reserve the term "trace class operator" for the special case of nuclear operators on Hilbert spaces, and reserve nuclear (=trace class) operators for more general Banach spaces.

Definition

Mimicking the definition for matrices, a bounded linear operator "A" over a separable Hilbert space "H" is said to be in the trace class if for some (and hence all) orthonormal bases {"e"_"k"}_"k" of "H" the sum of positive terms:$sum\_\{k\}\; langle\; (A^*A)^\{1/2\}\; ,\; e\_k,\; e\_k\; angle$is finite. In this case, the sum :$sum\_\{k\}\; langle\; A\; e\_k,\; e\_k\; angle$is absolutely convergent and is independent of the choice of the orthonormal basis. This value is called the trace of "A", denoted by Tr("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 of a matrix.

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 possibly divergent sum:$sum\_\{k\}\; langle\; A\; e\_k,\; e\_k\; angle.$

Properties

If "A" is a non-negative self-adjoint, "A" is trace class if and only if Tr("A") < &infin;. Therefore a self adjoint operator "A" is trace class if and only if its positive part "A"⁺ and negative part "A"⁻ are both trace class. (The positive and negative parts of a self adjoint operator are obtained via the continuous functional calculus.)

The trace is a linear functional over the space of trace class operators, i.e. :$operatorname\{Tr\}(aA+bB)=a,operatorname\{Tr\}(A)+b,operatorname\{Tr\}(B).$

It is continuous in the weak operator topology and satisfies properties 1 and 2 above.

The bilinear
$langle\; A,\; B\; angle\; =\; operatorname\{Tr\}(A^*\; B)$is an inner product on the trace class; the corresponding norm is called the Hilbert-Schmidt norm. The completion of the trace class operators in the Hilbert-Schmidt norm are called the Hilbert-Schmidt operators.

Relationship between some classes of operators

One can view certain classes of bounded operators as noncommutative analogue of classical sequence spaces. Trace-class operators as noncommutative analogue of the sequence space "l"¹(N). Indeed, applying the spectral theorem, every normal trace-class operator on a separable Hilbert space can be realized as a "l"¹ sequence. In the same vein, the bounded operators are noncommutative versions of "l"^&infin;(N), the compact operators that of "c"₀ (the sequences convergent to 0), Hilbert-Schmidt operators correspond to "l"²(N), and finite rank operators 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 "T" on a Hilbert space takes the following canonical form

:$forall\; h\; in\; H,\; ;\; T\; h\; =\; sum\; \_\{i\; =\; 1\}\; alpha\_i\; langle\; h,\; v\_i\; angle\; u\_i\; quad\; mbox\{where\}\; quad\; alpha\_i\; geq\; 0\; quad\; mbox\{and\}\; quad\; alpha\_i\; ightarrow\; 0$

for some orthonormal bases {"u_i"} and {"v_i"}. Making the above heuristic comments more precise, we have that "T" is trace class if the series &sum;_"i" "α_i" is convergent, "T" is Hilbert-Schmidt if &sum;_"i" "α_i"² is convergent, and "T" is finite rank if the sequence{"α_i"} has only finitely many nonzero terms.

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} &sub; {trace class} &sub; {Hilbert-Schmidt} &sub; {compact}.

The trace-class operators are given the trace norm ||"T"||₁ = Tr [ ("T*T")^&frac12; ] = &sum;_"i" "α_i". The norm corresponding to the Hilbert-Schmidt inner product is ||"T"||₂ = (Tr "T*T")^&frac12; = (&sum;_"i""α_i"²)^&frac12;. Also, the usual operator norm is ||"T"|| = sup_"i"("α_i"). By classical inequalities regarding sequences,

:$|T|\; leq\; |T|\_2\; leq\; |T|\_1\; ,$

for appropriate "T".

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 "c"₀ is "l"¹(N). Similarly, we have that the dual of compact operators, denoted by "K"("H")*, is the trace-class operators, denoted by "C"₁. The argument, which we now sketch, is reminiscent of that for the corresponding sequence spaces. Let "f" &isin; "K"("H")*, we identify "f" with the operator "T_f" defined by

:$langle\; T\_f\; x,\; y\; angle\; =\; f(S\_\{x,y\}),$

where "S"_"x,y" is the rank-one operator given by

:$S\_\{x,y\}(h)\; =\; langle\; h,\; y\; angle\; x.$

This identification works because the finite rank operators are norm-dense in "K"("H"). In the event that "T_f" is a positive operator, for any orthonormal basis "u_i", one has

:$sum\_i\; langle\; T\_f\; u\_i,\; u\_i\; angle\; =\; f(I)\; leq\; |f|,$

where "I" is the identity operator

:$I\; =\; sum\_i\; langle\; cdot,\; u\_i\; angle\; u\_i\; .$

But this means "T_f" is trace-class. An appeal to polar decomposition extend this to the general case where "T_f" need not be positive.

A limiting argument via finite rank operators shows that ||"T_f" ||₁ = || "f" ||. Thus "K"("H")* is isometrically isomorphic to "C"₁.

As the predual of bounded operators

Recall that the dual of "l"¹(N) is "l"^&infin;(N). In the present context, the dual of trace-class operators "C"₁ is the bounded operators B("H"). More precisely The set "C"₁ is a two-sided ideal in B("H"). So given any operator "T" in B("H"), we may define a continuous linear functional &phi;_"T" on $C\_1$ by &phi;_"T"("A")=Tr("AT"). This correspondence between elements &phi;_"T" of the dual space of $C\_1$ and bounded linear operators is an isometric isomorphism. It follows that B("H") "is" the dual space of $C\_1$. This can be used to defined the weak-* topology on B("H").

References

#Dixmier, J. (1969). "Les Algebres d'Operateurs dans l'Espace Hilbertien". Gauthier-Villars.