what does $\|X\|_{tr}$ stand for?

Question

what does $\|X\|_{tr}$ stand for ?
I meet this sign in my textbook but do no realize what it stands for.
Is it the same with $\|X\|_F$ ?
$\|X\|_F = \sqrt {\sum \limits_{i,j} X_{ij}^2}$

Answer 1

The notation $\|\cdot\|_{tr}$ should refer to the so called "trace-norm" which is the "Schatten p-norm" with $p = 1$ (see e.g. https://en.wikipedia.org/wiki/Schatten_norm).

Hence it is in general not equal to the Frobenius norm, allthough it is related to it, as the "Frobenius-norm" is the "Schatten p-norm" with $p = 2$.