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 μi(T) are the eigenvalues of |T| arranged in decreasing order. Let

.

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

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

  • limω(αn) ≥ 0 if all αn ≥ 0 (positivity)
  • limω(αn) = lim(αn) whenever the ordinary limit exists
  • limω(α1, α1, α2, α2, α3, ...) = limω(αn) (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 limω then the operator is called measurable.

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