Grothendieck trace theorem

Key takeaways

Grothendieck trace theorem belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Grothendieck trace theorem to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Grothendieck trace theorem from memory before moving on to harder problems.

Reference excerpt

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

2/3-nuclear operators Let A {\displaystyle A} be a nuclear operator on B {\displaystyle B} , then A {\displaystyle A} is a 2 3 {\displaystyle {\tfrac {2}{3}}} -nuclear operator if it has a decomposition of the form

A = ∑ k = 1 ∞ φ k ⊗ f k {\displaystyle A=\sum \limits _{k=1}^{\infty }\varphi _{k}\otimes f_{k}}

where φ k ∈ B {\displaystyle \varphi _{k}\in B} and f k ∈ B ′ {\displaystyle f_{k}\in B'} and

∑ k = 1 ∞ ‖ φ k ‖ 2 / 3 ‖ f k ‖ 2 / 3 < ∞ . {\displaystyle \sum \limits _{k=1}^{\infty }\|\varphi _{k}\|^{2/3}\|f_{k}\|^{2/3}<\infty .}

Grothendieck's trace theorem Let λ j ( A ) {\displaystyle \lambda _{j}(A)} denote the eigenvalues of a 2 3 {\displaystyle {\tfrac {2}{3}}} -nuclear operator A {\displaystyle A} counted with their algebraic multiplicities. If

∑ j | λ j ( A ) | < ∞ {\displaystyle \sum \limits _{j}|\lambda _{j}(A)|<\infty }

then the following equalities hold:

tr ⁡ A = ∑ j | λ j ( A ) | {\displaystyle \operatorname {tr} A=\sum \limits _{j}|\lambda _{j}(A)|}

and for the Fredholm determinant

det ⁡ ( I + A ) = ∏ j ( 1 + λ j ( A ) ) . {\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.

References

Worked examples

Example 1 — a first encounter with Grothendieck trace theorem

Start with the simplest possible case. Write down what Grothendieck trace theorem claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Grothendieck trace theorem before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Grothendieck trace theorem ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Grothendieck trace theorem

In research: Grothendieck trace theorem appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Grothendieck trace theorem in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Grothendieck trace theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Determinants, Theorems in functional analysis, Topological tensor products, so understanding it makes those chapters shorter.
In everyday life: Look for Grothendieck trace theorem outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Grothendieck trace theorem in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Grothendieck trace theorem means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Grothendieck trace theorem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Grothendieck trace theorem in simple terms?

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 2 3 {\displaystyle {\tfrac {2}{3}}} -nuclear operators. The theorem was proven in 1955 by Alexander…

Why does Grothendieck trace theorem matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Grothendieck trace theorem?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Grothendieck trace theorem.