The rectangular Bregman objective attains its minimum on the Schatten power sphere

Let $E,F$ be finite-dimensional complex inner-product spaces, let $\iota$ be a nonempty finite index type, let $p\in\mathbb R$ with $p>1$, and let $a:\iota\to\mathbb R$ (no sign condition on $a$). For a complex-linear map $T:E\to F$ with singular values $\operatorname{sv}_k(T)$, the Schatten $p$-power sphere is $\{T\mid\sum_k\operatorname{sv}_k(T)^p=1\}$ (schattenPowerSphere), and let $u:\iota\to$ (that sphere).

For $T:E\to F$, write $\widehat T$ for its Hermitian dilation, the self-adjoint operator on $E\oplus F$ with $\widehat T(x,y)=(T^\ast y,\,Tx)$ (hermitianDilation). For a self-adjoint operator $C$ and $h:\mathbb R\to\mathbb R$, write $h(C)$ for the result of applying $h$ to $C$'s eigenvalues in any orthonormal eigenbasis of $C$; this operator itself does not depend on which eigenbasis is chosen, since it acts as $h(\mu)$ on every $\mu$-eigenvector of $C$, not only on the vectors of one particular chosen eigenbasis — which is what lets $\widehat S\circ G_p(\widehat T)$ and $\widehat T\circ G_p(\widehat T)$ below be single well-defined operators. For $S,T:E\to F$, the dilated Bregman trace is

(dilatedBregmanTrace), with $F_p(x)=|x|^p/p$ and $G_p(x)=|x|^{p-2}x$. The rectangular Bregman objective is the half-dilated weighted sum

as a function of $v$ ranging over the Schatten $p$-power sphere, and $\mathrm{rectangularWeightedSum}(a,u)=\sum_i a_i\,u_i$ is the (unnormalized) weighted sum of the underlying maps. Writing $\|T\|_p=\big(\sum_k\operatorname{sv}_k(T)^p\big)^{1/p}$ for the Schatten $p$-norm (schattenPNorm), the theorem states that $\mathrm{rectangularBregmanObjective}(p,a,u,\cdot)$ attains the global minimum value

over the Schatten $p$-power sphere.

This is the operator-level analogue, for the Schatten $p$-norm on rectangular maps, of the exact Hilbert-space minimum of a weighted squared-distance objective: it is one of the two variational facts (the other being the analogous minimum for the paired Mazur-distance objective) whose comparison produces the two-sided bound between triple and pair deficits that this construction needs.

Formalization Note The Schatten $p$-power sphere is defined by $\sum_k\operatorname{sv}_k(T)^p=1$; for $p>0$ this is equivalent to $\|T\|_p=1$ (schattenPNorm_eq_one_iff). In this theorem's range $p>1$, $F_p$ is convex and differentiable with derivative $G_p$. The proof uses the resulting scalar Bregman nonnegativity through a supporting theorem, then lifts it to $B_p$ by the spectral trace identity.