Dixmier Trace - Definition

Definition

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

is finite, where the numbers μ_i(T) are the eigenvalues of |T| arranged in decreasing order. Let

.

The Dixmier trace Tr_ω(T) of T is defined for positive operators T of L1,∞(H) to be

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(α₁, α₁, α₂, α₂, α₃, ...) = lim_ω(α_n) (scale invariance)

There are many such extensions (such as a Banach limit of α₁, α₂, α₄, 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 lim_ω then the operator is called measurable.