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.
[1] The theorem was proven in 1955 by Alexander Grothendieck.
[2] 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.
Given a Banach space
with the approximation property and denote its dual as
be a nuclear operator on
-nuclear operator if it has a decomposition of the form
λ
denote the eigenvalues of a
-nuclear operator
counted with their algebraic multiplicities.
λ
then the following equalities hold:
tr
λ
{\displaystyle \operatorname {tr} A=\sum \limits _{j}|\lambda _{j}(A)|}
and for the Fredholm determinant
det (
λ
{\displaystyle \operatorname {det} (I+A)=\prod \limits _{j}(1+\lambda _{j}(A)).}