Suppose $T$ maps between symmetric positive definite $d$-by-$d$ matrices $X,Y$ as follows:
$$\operatorname{vec}(Y)=M\operatorname{vec}(X)$$
$M$ is a symmetric positive definite $d^2$-by-$d^2$ matrix. Suppose we define norm $\|T\|_*$ to be the spectral norm of $M$, is there another name for $\|\cdot\|_*$?
This norm comes up indirectly in statistics, and I was curious to find another representation for it. A related example uses Russo-Dye to give a simple expression for induced trace norm $\|T\|_1$, page 167 of Theory of Quantum Information textbook