Grothendieck trace theorem

In functional analysis, the Grothendieck trace theorem is an extension of Lidskii's theorem about the trace and the determinant of a certain class of nuclear operators on Banach spaces, the so-called -nuclear operators. The theorem was proven in 1955 by Alexander Grothendieck.* Lidskii's theorem does not hold in general for Banach spaces.

The theorem should not be confused with the Grothendieck trace formula from algebraic geometry.

Grothendieck trace theorem

Given a Banach space with the approximation property and denote its dual as .

2/3-nuclear operators

Let be a nuclear operator on , then is a -nuclear operator if it has a decomposition of the form

where and and

Grothendieck's trace theorem

Let denote the eigenvalues of a -nuclear operator counted with their algebraic multiplicities. If

then the following equalities hold:

and for the Fredholm determinant

See also

Literature

References