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 form$$A\; =\; \backslash sum\backslash limits\_^\backslash varphi\_k\; \backslash otimes\; f\_k$$where \varphi _{k} \in B and f _{k} \in B' and

Grothendieck's trace theorem

Let

λ _{j(A)} denote the eigenvalues of a \tfrac{ 2 }{ 3 } -nuclear operator A counted with their algebraic multiplicities. Ifthen the following equalities hold:$$\backslash operatornameA\; =\; \backslash sum\backslash limits\_j\; |\backslash lambda\_j(A)|$$and for the Fredholm determinant$$\backslash operatorname(I+A)\; =\; \backslash prod\backslash limits\_j\; (1+\backslash lambda\_j(A)).$$

Literature

• 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

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.
2. • Book: Grothendieck, Alexander . Produits tensoriels topologiques et espaces nucléaires . American Mathematical Society . Providence . 19. 1955. 0-8218-1216-5 . 1315788 . fr.