Triple-deficit upper bound for the Schatten $p$-norm via the radial Mazur map

Let $E,F$ be finite-dimensional complex inner-product spaces, let $p>1$, let $m,M\in\mathbb{R}$ with $m\ge0$, and let $x,y,z:E\to F$ be any three complex-linear maps, possibly zero. Write $\|\cdot\|_p$ for the Schatten $p$-norm (schattenPNorm). The rectangular Mazur map $M_p$ (rectangularMazurMap) is obtained by applying $t\mapsto\operatorname{sign}(t)|t|^{p/2}$ to the eigenvalues of the Hermitian dilation $\widehat T(\xi,\eta)=(T^\ast\eta,T\xi)$ on $E\oplus F$, then taking the lower-left block. It carries the Schatten-$p$ power sphere onto the Schatten-$2$ power sphere. It is generally nonlinear; at $p=2$ it is the identity. Define the radial rectangular image $\Phi_p$ (radialRectangularMazurMap) by

Record $\Phi_p(T)$ by its values on a fixed orthonormal basis of $E$, obtaining the vector $\Psi_p(T)$ in the Hilbert space of column tuples with its $\ell^2$ norm (radialMazurHilbertMap). Then $\Psi_p(0)=0$ and $\|\Psi_p(T)\|=\|T\|_p$ for every $T$. Define the triple deficit and its mapped counterpart by

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 between $\widehat S$ and $\widehat T$, built respectively from the scalar potential $|t|^p/p$ and the odd map $t\mapsto\operatorname{sign}(t)|t|^{p/2}$. Assume the uniform two-sided bound

holds for every pair $S,T:E\to F$. Then

This is the three-operator specialization of the family-level deficit estimate, extended to include vanishing inputs. Only the upper bound needed in the final transfer is stated. When $m>0$ and $M\ge0$, it combines with the three pair lower bounds and the classical Hilbert-space Hlawka inequality for the mapped vectors to give a Hlawka constant $M/m$. If the same comparison constants work in every dimension, the resulting constant is dimension independent.