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 $\frac{2}{3}$ -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 $(B, ‖ \cdot ‖)$ 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 $\frac{2}{3}$ -nuclear operator if it has a decomposition of the form $A = \sum_{k = 1}^{\infty} φ_{k} \otimes f_{k}$ where $φ_{k} \in B$ and $f_{k} \in B^{'}$ and $\sum_{k = 1}^{\infty} ‖ φ_{k} ‖^{2 / 3} ‖ f_{k} ‖^{2 / 3} < \infty .$

Grothendieck's trace theorem

Let $λ_{j} (A)$ denote the eigenvalues of a $\frac{2}{3}$ -nuclear operator $A$ counted with their algebraic multiplicities. If $\sum_{j} | λ_{j} (A) | < \infty$ then the following equalities hold: $tr A = \sum_{j} | λ_{j} (A) |$ and for the Fredholm determinant $\det (I + A) = \prod_{j} (1 + λ_{j} (A)) .$

Literature

Gohberg, Israel; Goldberg, Seymour; Krupnik, Nahum (1991). Traces and Determinants of Linear Operators. Operator Theory Advances and Applications. Basel: Birkhäuser. p. 102. ISBN 978-3-7643-6177-8.

References

↑ Gohberg, Israel; Goldberg, Seymour; Krupnik, Nahum (1991). Traces and Determinants of Linear Operators. Operator Theory Advances and Applications. Basel: Birkhäuser. p. 102. ISBN 978-3-7643-6177-8.
↑ * Grothendieck, Alexander (1955). Produits tensoriels topologiques et espaces nucléaires (in français). Providence: American Mathematical Society. p. 19. ISBN 0-8218-1216-5. OCLC 1315788.

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Grothendieck trace theorem

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