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 $A$ 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.

Topological tensor products and nuclear spaces
Basic concepts	Auxiliary normed spaces Nuclear space Tensor product Topological tensor product of Hilbert spaces
Topologies	Inductive tensor product Injective tensor product Projective tensor product
Operators/Maps	Fredholm determinant Fredholm kernel Hilbert–Schmidt operator Hypocontinuity Integral Nuclear between Banach spaces Trace class
Theorems	Grothendieck trace theorem Schwartz kernel theorem

Banach space topics
Types of Banach spaces	Asplund Banach list Banach lattice Grothendieck Hilbert Inner product space Polarization identity (Polynomially) Reflexive Riesz L-semi-inner product (B Strictly Uniformly) convex Uniformly smooth (Injective Projective) Tensor product (of Hilbert spaces)
Banach spaces are:	Barrelled Complete F-space Fréchet tame Locally convex Seminorms/Minkowski functionals Mackey Metrizable Normed norm Quasinormed Stereotype
Function space Topologies	Banach–Mazur compactum Dual Dual space Dual norm Operator Ultraweak Weak polar operator Strong polar operator Ultrastrong Uniform convergence
Linear operators	Adjoint Bilinear form operator sesquilinear (Un)Bounded Closed Compact on Hilbert spaces (Dis)Continuous Densely defined Fredholm kernel operator Hilbert–Schmidt Functionals positive Pseudo-monotone Normal Nuclear Self-adjoint Strictly singular Trace class Transpose Unitary
Operator theory	Banach algebras C-algebras Operator space Spectrum C-algebra radius Spectral theory of ODEs Spectral theorem Polar decomposition Singular value decomposition
Theorems	Anderson–Kadec Banach–Alaoglu Banach–Mazur Banach–Saks Banach–Schauder (open mapping) Banach–Steinhaus (Uniform boundedness) Bessel's inequality Cauchy–Schwarz inequality Closed graph Closed range Eberlein–Šmulian Freudenthal spectral Gelfand–Mazur Gelfand–Naimark Goldstine Hahn–Banach hyperplane separation Kakutani fixed-point Krein–Milman Lomonosov's invariant subspace Mackey–Arens Mazur's lemma M. Riesz extension Parseval's identity Riesz's lemma Riesz representation Robinson-Ursescu Schauder fixed-point
Analysis	Abstract Wiener space Banach manifold bundle Bochner space Convex series Differentiation in Fréchet spaces Derivatives Fréchet Gateaux functional holomorphic quasi Integrals Bochner Dunford Gelfand–Pettis regulated Paley–Wiener weak Functional calculus Borel continuous holomorphic Measures Lebesgue Projection-valued Vector Weakly / Strongly measurable function
Types of sets	Absolutely convex Absorbing Affine Balanced/Circled Bounded Convex Convex cone (subset) Convex series related ((cs, lcs)-closed, (cs, bcs)-complete, (lower) ideally convex, (Hx), and (Hwx)) Linear cone (subset) Radial Radially convex/Star-shaped Symmetric Zonotope
Subsets / set operations	Affine hull (Relative) Algebraic interior (core) Bounding points Convex hull Extreme point Interior Linear span Minkowski addition Polar (Quasi) Relative interior
Examples	Absolute continuity AC $ba(\Sigma )$ c space Banach coordinate BK Besov $B_{p,q}^{s}(\mathbb {R} )$ Birnbaum–Orlicz Bounded variation BV Bs space Continuous C(K) with K compact Hausdorff Hardy H^p Hilbert H Morrey–Campanato $L^{\lambda ,p}(\Omega )$ ℓ^p $\ell ^{\infty }$ L^p $L^{\infty }$ weighted Schwartz $S\left(\mathbb {R} ^{n}\right)$ Segal–Bargmann F Sequence space Sobolev W^k,p Sobolev inequality Triebel–Lizorkin Wiener amalgam $W(X,L^{p})$
Applications	Differential operator Finite element method Mathematical formulation of quantum mechanics Ordinary Differential Equations (ODEs) Validated numerics

Functional analysis (topics – glossary)

Spaces

Banach Besov Fréchet Hilbert Hölder Nuclear Orlicz Schwartz Sobolev Topological vector
Properties	Barrelled Complete Dual (Algebraic/Topological) Locally convex Reflexive Separable

Theorems

Operators

Algebras

Open problems

Applications

Advanced topics

Mathematics portal
Category
Commons

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

Grothendieck trace theorem