Trace class

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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[edit]

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 {ek}k of H the sum of positive terms

\|A\|_{1}= {\rm Tr}|A|:=\sum_{k} \langle (A^*A)^{1/2} \, e_k, e_k \rangle

is finite. In this case, the sum

{\rm Tr} A:=\sum_{k} \langle A e_k, e_k \rangle

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 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 \rangle.

Properties[edit]

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+ 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.)
2. 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).

The bilinear map

 \langle A, B \rangle = \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.

3. If A is bounded and B is trace class, AB and BA are also trace class and [1]
 \|AB\|_1=\operatorname{Tr}(|AB|)\le \|A\|\|B\|_1,\qquad \|BA\|_1=\operatorname{Tr}(|BA|)\le \|A\|\|B\|_1

besides, under the same hypothesis,

 \operatorname{Tr}(AB)=\operatorname{Tr}(BA)

The last assertion also holds under the weaker hypothesis that A and B are Hilbert Schmidt.

4. If A is trace class, then one can define the Fredholm determinant of 1+A
 {\rm det} (I+A):=\prod_{n\ge 1}[1+\lambda_n(A)]

for \{\lambda_n(A)\}_n the elements of the spectrum of A; the trace class condition on A guarantees that the infinite product is finite: indeed

 {\rm det} (I+A)\le e^{\|A\|_1};

it also guarantees that  {\rm det} (I+A)\neq 0 if and only if  (I+A) is invertible.

Lidskii's theorem[edit]

Let A be a trace class operator in a separable Hilbert space H, and let \{\lambda_n(A)\}_{n=1}^N, N\leq \infty be the eigenvalues of A. Let us assume that \lambda_n(A) are enumerated with algebraic multiplicities taken into account (i.e. if the algebraic multiplicity of \lambda is k then \lambda is repeated k times in the list \lambda_1(A),\lambda_2(A),\dots). Lidskii's theorem (named after Victor Borisovich Lidskii) states that

\sum_{n=1}^N \lambda_n(A)=\operatorname{Tr}(A).

Note that the series in the left hand side converges absolutely due to Weyl's inequality

\sum_{n=1}^N |\lambda_n(A)|\leq \sum_{m=1}^M s_m(A)

between the eigenvalues \{\lambda_n(A)\}_{n=1}^N and the singular values \{s_m(A)\}_{m=1}^M of a compact operator A. See e.g.[2]

Relationship between some classes of operators[edit]

One can view certain classes of bounded operators as noncommutative analogue of classical sequence spaces, with trace-class operators as the noncommutative analogue of the sequence space l1(N). Indeed, applying the spectral theorem, every normal trace-class operator on a separable Hilbert space can be realized as a l1 sequence. In the same vein, the bounded operators are noncommutative versions of l(N), the compact operators that of c0 (the sequences convergent to 0), Hilbert-Schmidt operators correspond to l2(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\rangle u_i \quad \mbox{where} \quad \alpha_i \geq 0 \quad \mbox{and} \quad \alpha_i \rightarrow 0

for some orthonormal bases {ui} and {vi}. Making the above heuristic comments more precise, we have that T is trace class if the series ∑i αi is convergent, T is Hilbert-Schmidt if ∑i αi2 is convergent, and T is finite rank if the sequence {αi} 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} ⊂ {trace class} ⊂ {Hilbert-Schmidt} ⊂ {compact}.

The trace-class operators are given the trace norm ||T||1 = Tr [ (T*T)½ ] = ∑i αi. The norm corresponding to the Hilbert-Schmidt inner product is ||T||2 = (Tr T*T)½ = (∑iαi2)½. Also, the usual operator norm is ||T|| = supi(αi). 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[edit]

The dual space of c0 is l1(N). Similarly, we have that the dual of compact operators, denoted by K(H)*, is the trace-class operators, denoted by C1. The argument, which we now sketch, is reminiscent of that for the corresponding sequence spaces. Let fK(H)*, we identify f with the operator Tf defined by

\langle T_f x, y \rangle = f(S_{x,y}),

where Sx,y is the rank-one operator given by

S_{x,y}(h) = \langle h, y \rangle x.

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

\sum_i \langle T_f u_i, u_i \rangle = f(I) \leq \|f\|,

where I is the identity operator

I = \sum_i \langle \cdot, u_i \rangle u_i .

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

A limiting argument via finite-rank operators shows that ||Tf ||1 = || f ||. Thus K(H)* is isometrically isomorphic to C1.

As the predual of bounded operators[edit]

Recall that the dual of l1(N) is l(N). In the present context, the dual of trace-class operators C1 is the bounded operators B(H). More precisely, the set C1 is a two-sided ideal in B(H). So given any operator T in B(H), we may define a continuous linear functional φT on C_1 by φT(A)=Tr(AT). This correspondence between elements φT 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 define the weak-* topology on B(H).

Notes[edit]

  1. ^ M. Reed and B. Simon Functional Analysis, Exercises 27, 28 page 218
  2. ^ Simon, B. (2005) Trace ideals and their applications, Second Edition, Amer. Math. Soc.

References[edit]

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