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 ${\tfrac {2}{3}}$ -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 $(B,\|\cdot \|)$ 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 ${\tfrac {2}{3}}$ -nuclear operator if it has a decomposition of the form

A=\sum \limits _{k=1}^{\infty }\varphi _{k}\otimes f_{k}

where $\varphi _{k}\in B$ and $f_{k}\in B'$ and

\sum \limits _{k=1}^{\infty }\|\varphi _{k}\|^{2/3}\|f_{k}\|^{2/3}<\infty .

Grothendieck's trace theorem

Let $\lambda _{j}(A)$ denote the eigenvalues of a ${\tfrac {2}{3}}$ -nuclear operator $A$ counted with their algebraic multiplicities. If

\sum \limits _{j}|\lambda _{j}(A)|<\infty

then the following equalities hold:

\operatorname {tr} A=\sum \limits _{j}|\lambda _{j}(A)|

and for the Fredholm determinant

\operatorname {det} (I+A)=\prod \limits _{j}(1+\lambda _{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 French). Providence: American Mathematical Society. p. 19. ISBN 0-8218-1216-5. OCLC 1315788.

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