Norm ideal

This is a draft article. It is a work in progress open to editing by anyone. Please ensure core content policies are met before publishing it as a live Wikipedia article.

Find sources: Google (books · news · scholar · free images · WP refs) · FENS · JSTOR · TWL
Easy tools: Citation bot (help) | Advanced: Fix bare URLs · Article logs · Draft logs.

Last edited by Cosmia Nebula (talk | contribs) 27 hours ago. (Update)

Finished drafting? Submit for review

In mathematics, especially functional analysis, a norm ideal is a specific kind of ideal in the algebra of operators over a Hilbert space.

Let ${\textstyle {\mathcal {H}}}$ be a Hilbert space. Let ${\textstyle {\mathcal {B}}({\mathcal {H}})}$ be the Banach algebra of bounded operators over ${\textstyle {\mathcal {H}}}$ .

A norm ideal is a two-sided ideal ${\textstyle {\mathcal {C}}}$ in ${\textstyle {\mathcal {B}}({\mathcal {H}})}$ equipped with a norm ${\textstyle \|.\|_{\mathcal {C}}}$ which has the following properties:^[1]

For any ${\textstyle S,T\in {\mathcal {B}}({\mathcal {H}})}$ and ${\textstyle A\in {\mathcal {C}},\|SAT\|_{\mathcal {C}}\leq \|S\|\|A\|_{\mathcal {C}}\|T\|}$ .
If ${\textstyle A\in {\mathcal {C}}}$ , then ${\textstyle A^{*}\in {\mathcal {C}}}$ and ${\textstyle \left\|A^{*}\right\|_{\mathcal {C}}=\|A\|_{\mathcal {C}}}$ .
For any ${\textstyle A\in {\mathcal {C}},\|A\|\leq \|A\|_{\mathcal {C}}}$ , and the equality holds when ${\textstyle \operatorname {rank} (A)=1}$ .
${\textstyle {\mathcal {C}}}$ is complete with respect to ${\textstyle \|\cdot \|_{\mathcal {C}}}$ .
${\textstyle {\mathcal {C}}\neq \{0\}}$ .

The most important examples are the p-Schatten classes with p-Schatten norms. The $p=1$ case is the trace class. The $p=2$ case is the Hilbert–Schmidt class.

References

Schatten, Robert (1960). Norm Ideals of Completely Continuous Operators. Ergebnisse der Mathematik und ihrer Grenzgebiete. Berlin: Springer-Verlag.

Spectral theory and ^*-algebras
Basic concepts	Involution/-algebra Banach algebra B-algebra C-algebra Noncommutative topology Projection-valued measure Spectrum Spectrum of a C-algebra Spectral radius Operator space
Main results	Gelfand–Mazur theorem Gelfand–Naimark theorem Gelfand representation Polar decomposition Singular value decomposition Spectral theorem Spectral theory of normal C*-algebras
Special Elements/Operators	Isospectral Normal operator Hermitian/Self-adjoint operator Unitary operator Unit
Spectrum	Krein–Rutman theorem Normal eigenvalue Spectrum of a C*-algebra Spectral radius Spectral asymmetry Spectral gap
Decomposition	Decomposition of a spectrum Continuous Point Residual Approximate point Compression Direct integral Discrete Spectral abscissa
Spectral Theorem	Borel functional calculus Min-max theorem Positive operator-valued measure Projection-valued measure Riesz projector Rigged Hilbert space Spectral theorem Spectral theory of compact operators Spectral theory of normal C*-algebras
Special algebras	Amenable Banach algebra With an Approximate identity Banach function algebra Disk algebra Nuclear C*-algebra Uniform algebra Von Neumann algebra Tomita–Takesaki theory
Finite-Dimensional	Alon–Boppana bound Bauer–Fike theorem Numerical range Schur–Horn theorem
Generalizations	Dirac spectrum Essential spectrum Pseudospectrum Structure space (Shilov boundary)
Miscellaneous	Abstract index group Banach algebra cohomology Cohen–Hewitt factorization theorem Extensions of symmetric operators Fredholm theory Limiting absorption principle Schröder–Bernstein theorems for operator algebras Sherman–Takeda theorem Unbounded operator
Examples	Wiener algebra
Applications	Almost Mathieu operator Corona theorem Hearing the shape of a drum (Dirichlet eigenvalue) Heat kernel Kuznetsov trace formula Lax pair Proto-value function Ramanujan graph Rayleigh–Faber–Krahn inequality Spectral geometry Spectral method Spectral theory of ordinary differential equations Sturm–Liouville theory Superstrong approximation Transfer operator Transform theory Weyl law Wiener–Khinchin theorem

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

Category

This page will be placed in the following categories if it is moved to the article namespace.

Categories:

Linear operators
Operator theory

^ (Schatten 1960, p. vi)