Norm ideal

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

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

A norm ideal is a two-sided ideal ${𝒞}$ in ${ℬ}({\mathscr{H}})$ equipped with a norm $\Vert .{\Vert }_{{𝒞}}$ which has the following properties:

• For any $S,T\in {ℬ}({\mathscr{H}})$ and $A\in {𝒞},\Vert SAT{\Vert }_{{𝒞}}\le \Vert S\Vert \Vert A{\Vert }_{{𝒞}}\Vert T\Vert$.
• If $A\in {𝒞}$, then $A^{*}\in {𝒞}$ and ${\Vert A^{*}\Vert }_{{𝒞}}=\Vert A{\Vert }_{{𝒞}}$.
• For any $A\in {𝒞},\Vert A\Vert \le \Vert A{\Vert }_{{𝒞}}$, and the equality holds when $\mathrm{rank}(A)=1$.
• ${𝒞}$ is complete with respect to $\Vert \cdot {\Vert }_{{𝒞}}$.
• ${𝒞}\ne \{0\}$.

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. Search this book on

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