2-summing operator (${\pi }_2(T)$)

A bounded linear operator $T:E\to F$ is 2-summing if ${\sum }_j\Vert T{x}_j{\Vert }^2\le C^2{\sup }_{\varphi \in B_{E^{\ast }}}{\sum }_j|\varphi (x_j){|}^2$ for every finite family; ${\pi }_2(T)$ is the least $C$. For operators between Hilbert spaces, this is equivalent to being a Hilbert-Schmidt operator, with ${\pi }_2(T)=\Vert T{\Vert }_{\mathrm{HS}}$. The upper bound follows by applying the Hilbert-Schmidt norm ideal inequality to the column operator of the family; testing finite orthonormal families gives the reverse bound.

For a finite-dimensional normed vector space $E$, represent a finite family by its column operator $A:{\ell }_2^m\to E$. Its operator norm is the family's weak square norm. Since $A=AQ$ for the orthogonal projection onto $(\ker A{)}^{\perp }$, summing $\Vert A{e}_j{\Vert }^2\le \Vert A{\Vert }^2\Vert Q{e}_j{\Vert }_2^2$ proves the upper bound $\sqrt{\dim E}$. In coordinates adapted to a maximal inscribed Euclidean or Hermitian ellipsoid, the John contact decomposition gives ${\sum }_r{c}_r{u}_r{u}_r^{\ast }=I$ and ${\sum }_r{c}_r=\dim E$. The vectors $\sqrt{c_r}{u}_r$ have total squared norm $\dim E$ and weak square norm one, giving equality. This holds over both real and complex scalars.