Grothendieck trace theorem explained
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.
Grothendieck trace theorem
Given a Banach space
with the
approximation property and denote its
dual as
.
⅔-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 determinantLiterature
- Book: Israel. Gohberg. Seymour. Goldberg. Nahum. Krupnik. Traces and Determinants of Linear Operators. Operator Theory Advances and Applications . Birkhäuser. Basel. 1991. 978-3-7643-6177-8. 102.
Notes and References
- Book: Israel. Gohberg. Seymour. Goldberg. Nahum. Krupnik. Traces and Determinants of Linear Operators. Operator Theory Advances and Applications . Birkhäuser . Basel . 1991 . 978-3-7643-6177-8. 102.
- Book: Grothendieck, Alexander . Produits tensoriels topologiques et espaces nucléaires . American Mathematical Society . Providence . 19. 1955. 0-8218-1216-5 . 1315788 . fr.