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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