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

\tfrac{2}{3}

-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

(B,\|\|)

with the approximation property and denote its dual as

B'

.

⅔-nuclear operators

Let

A

be a nuclear operator on

B

, then

A

is a

\tfrac{2}{3}

-nuclear operator
if it has a decomposition of the formA = \sum\limits_^\varphi_k \otimes f_kwhere

\varphik\inB

and

fk\inB'

and\sum\limits_^\|\varphi_k\|^ \|f_k\|^ < \infty.

Grothendieck's trace theorem

Let

λj(A)

denote the eigenvalues of a

\tfrac{2}{3}

-nuclear operator

A

counted with their algebraic multiplicities. If\sum\limits_j |\lambda_j(A)| < \inftythen the following equalities hold:\operatornameA = \sum\limits_j |\lambda_j(A)|and for the Fredholm determinant\operatorname(I+A) = \prod\limits_j (1+\lambda_j(A)).

Literature

Notes and References

  1. 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.