Dixmier trace

Definition

If H is a Hilbert space, then L1,∞(H) is the space of compact linear operators T on H such that the norm

:|T|{1,\infty} = \sup_N\frac{\sum{i=1}^N \mu_i(T)}{\log(N)}

is finite, where the numbers μi(T) are the eigenvalues of |T| arranged in decreasing order. Let :a_N = \frac{\sum_{i=1}^N \mu_i(T)}{\log(N)}. The Dixmier trace Trω(T) of T is defined for positive operators T of L1,∞(H) to be

:\operatorname{Tr}\omega(T)= \lim\omega a_N

where lim*ω* is a scale-invariant positive "extension" of the usual limit, to all bounded sequences. In other words, it has the following properties:

• lim*ω(α*n) ≥ 0 if all *α*n ≥ 0 (positivity)
• lim*ω(α*n) = lim(*α*n) whenever the ordinary limit exists
• lim*ω(α*1, *α*1, *α*2, *α*2, *α3, ...) = limω(α*n) (scale invariance)

There are many such extensions (such as a Banach limit of *α*1, *α*2, *α*4, α8,...) so there are many different Dixmier traces. As the Dixmier trace is linear, it extends by linearity to all operators of L1,∞(H). If the Dixmier trace of an operator is independent of the choice of limω then the operator is called measurable.

Properties

• Tr*ω*(T) is linear in T.
• If T ≥ 0 then Tr*ω*(T) ≥ 0
• If S is bounded then Trω(ST) = Tr*ω*(TS)
• Trω(T) does not depend on the choice of inner product on H.
• Tr*ω*(T) = 0 for all trace class operators T, but there are compact operators for which it is equal to 1.

A trace φ is called normal if φ(sup xα) = sup φ( x*α*) for every bounded increasing directed family of positive operators. Any normal trace on L^{1,\infty}(H) is equal to the usual trace, so the Dixmier trace is an example of a non-normal trace.

Examples

A compact self-adjoint operator with eigenvalues 1, 1/2, 1/3, ... has Dixmier trace equal to 1.

If the eigenvalues μi of the positive operator T have the property that :\zeta_T(s)= \operatorname{Tr}(T^s)= \sum{\mu_i^s} converges for Re(s)1 and extends to a meromorphic function near s=1 with at most a simple pole at s=1, then the Dixmier trace of T is the residue at s=1 (and in particular is independent of the choice of ω).

showed that Wodzicki's noncommutative residue of a pseudodifferential operator on a manifold M of order -dim(M) is equal to its Dixmier trace.

References

• Albeverio, S.; Guido, D.; Ponosov, A.; Scarlatti, S.: Singular traces and compact operators. J. Funct. Anal. 137 (1996), no. 2, 281—302.