Extension of Lidskii's 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 formula_1-nuclear operators. The theorem was proven in 1955 by Alexander Grothendieck. 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 formula_2 with the approximation property and denote its dual as formula_3.
⅔-nuclear operators.
Let formula_4 be a nuclear operator on formula_5, then formula_4 is a "formula_1-nuclear operator" if it has a decomposition of the form
formula_8
where formula_9 and formula_10 and
formula_11
Grothendieck's trace theorem.
Let formula_12 denote the eigenvalues of a "formula_1-nuclear operator" formula_4 counted with their algebraic multiplicities. If
formula_15
then the following equalities hold:
formula_16
and for the Fredholm determinant
formula_17
References.
 ;templatestyles src="Reflist/styles.css" / ;