Norm ideal

In mathematics, especially functional analysis, a norm ideal, or a symmetrically normed 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:

• 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 ${\displaystyle p=1}$ case is the trace class. The ${\displaystyle p=2}$ 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.