Two-sided comparison of a Schatten family deficit with its radial Mazur image

Let $E,F$ be finite-dimensional complex inner-product spaces, let $\iota$ be a nonempty finite index set, let $p>1$, and let $m,M\in\mathbb{R}$ with $m\ge0$. Let $(x_i)_{i\in\iota}$ be a family of nonzero complex-linear maps $E\to F$. Write $\|\cdot\|_p$ for the Schatten $p$-norm and $\|\cdot\|_2$ for the Schatten-$2$ (Hilbert–Schmidt) norm (schattenPNorm). Let $M_p(T)$ be the rectangular Mazur map (rectangularMazurMap): apply the odd scalar map $t\mapsto\operatorname{sign}(t)|t|^{p/2}$ to the eigenvalues of the Hermitian dilation $\widehat T(\xi,\eta)=(T^\ast\eta,T\xi)$ on $E\oplus F$, then take the lower-left block. Define its radial version $\Phi_p$ (radialRectangularMazurMap) by

Thus $\|\Phi_p(T)\|_2=\|T\|_p$ for every $T$.

For any two complex-linear maps $S,T:E\to F$, write $B_p(S,T)$ (dilatedBregmanTrace) and $D_p(S,T)$ (dilatedMazurDistanceSq) for the trace-level Bregman divergence and squared Mazur distance, at exponent $p$, between the two Hermitian dilations $\widehat S$ and $\widehat T$ on $E\oplus F$ — built respectively from the scalar potential $|t|^p/p$ and the odd power map $t\mapsto\operatorname{sign}(t)|t|^{p/2}$. Suppose there is a uniform two-sided bound

holding for every pair of maps $S,T:E\to F$. Then

This bounds, on both sides, the actual multi-operator Schatten-$p$ deficit $\sum_i\|x_i\|_p-\|\sum_i x_i\|_p$ of a whole finite family by the corresponding deficit computed after mapping every operator, radially, into Hilbert–Schmidt space. Specializing $\iota$ to two or three elements produces the pair and triple deficit comparisons used later in the assembly of the dimension-independent Hlawka constant.

Formalization Note The quantity $\|\sum_i\Phi_p(x_i)\|_2$ on the two outer sides is the Schatten-$2$ norm of a sum of operators. Evaluating an operator on a fixed orthonormal basis of $E$ is linear, and the Schatten-$2$ norm of an operator is the Hilbert-space norm of that family of values (schattenPNorm_two_eq_norm_hilbertSchmidtCoordinates), so this quantity also equals $\|\sum_i\Psi_p(x_i)\|$, where $\Psi_p(x_i)$ (radialMazurHilbertMap p (x_i)) is $\Phi_p(x_i)$ recorded by its values on that basis.