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 ${\displaystyle {\tfrac {2}{3}}}$-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 ${\displaystyle (B,\|\cdot \|)}$ with the approximation property and denote its dual as ${\displaystyle B'}$.

2/3-nuclear operators

Let ${\displaystyle A}$ be a nuclear operator on ${\displaystyle B}$, then ${\displaystyle A}$ is a ${\displaystyle {\tfrac {2}{3}}}$-nuclear operator if it has a decomposition of the form $${\displaystyle A=\sum \limits _{k=1}^{\infty }\varphi _{k}\otimes f_{k}}$$ where ${\displaystyle \varphi _{k}\in B}$ and ${\displaystyle f_{k}\in B'}$ and $${\displaystyle \sum \limits _{k=1}^{\infty }\|\varphi _{k}\|^{2/3}\|f_{k}\|^{2/3}<\infty .}$$

Grothendieck's trace theorem

Let ${\displaystyle \lambda _{j}(A)}$ denote the eigenvalues of a ${\displaystyle {\tfrac {2}{3}}}$-nuclear operator ${\displaystyle A}$ counted with their algebraic multiplicities. If $${\displaystyle \sum \limits _{j}|\lambda _{j}(A)|<\infty }$$ then the following equalities hold: $${\displaystyle \operatorname {tr} A=\sum \limits _{j}|\lambda _{j}(A)|}$$ and for the Fredholm determinant $${\displaystyle \operatorname {det} (I+A)=\prod \limits _{j}(1+\lambda _{j}(A)).}$$

See also

• Nuclear operators between Banach spaces

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.