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.
Read more about this topic: Dixmier Trace
Famous quotes containing the word definition:
“It is very hard to give a just definition of love. The most we can say of it is this: that in the soul, it is a desire to rule; in the spirit, it is a sympathy; and in the body, it is but a hidden and subtle desire to possess—after many mysteries—what one loves.”
—François, Duc De La Rochefoucauld (1613–1680)
“Although there is no universal agreement as to a definition of life, its biological manifestations are generally considered to be organization, metabolism, growth, irritability, adaptation, and reproduction.”
—The Columbia Encyclopedia, Fifth Edition, the first sentence of the article on “life” (based on wording in the First Edition, 1935)
“Perhaps the best definition of progress would be the continuing efforts of men and women to narrow the gap between the convenience of the powers that be and the unwritten charter.”
—Nadine Gordimer (b. 1923)