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.{{cite book|first1=Israel|last1=Gohberg|first2=Seymour|last2=Goldberg|first3=Nahum|last3=Krupnik|title=Traces and Determinants of Linear Operators|series=Operator Theory Advances and Applications |publisher=Birkhäuser |place=Basel |date=1991 |isbn=978-3-7643-6177-8|page=102}} The theorem was proven in 1955 by Alexander Grothendieck.* {{cite book | last=Grothendieck | first=Alexander | title=Produits tensoriels topologiques et espaces nucléaires | publisher=American Mathematical Society | location=Providence | page=19| year=1955| isbn=0-8218-1216-5 | oclc=1315788 | language=fr}} 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 $B'$ .

=2/3-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

{{cite book|first1=Israel|last1=Gohberg|first2=Seymour|last2=Goldberg|first3=Nahum|last3=Krupnik|title=Traces and Determinants of Linear Operators|series=Operator Theory Advances and Applications |publisher=Birkhäuser|place=Basel|date=1991|isbn=978-3-7643-6177-8|page=102}}

References

Category:Theorems in functional analysis

Category:Topological tensor products

Category:Determinants

Grothendieck trace theorem