I'd like some help showing the equivalence of these two norms when $p = \log n$.
Recall the $p$-th Schatten norm of a linear operator $A$ acting on $\mathbb{R}^{n}$. In the particular case of $p = \log(n)$, Schatten norm should be equivalent to the spectral norm, see last line on p.18. Moreover, $\| A\|_{C^{n}_{\log n}} \leq \| A \|$.
On the other hand, consider an example of an operator on $\mathbb{R}^{3}$:
$ A = \begin{pmatrix} 1 & 0 & 0\\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{pmatrix} $
Obviously, $\| A \| = 3$. However when $p = \log(3)$, the corresponding Schatten's norm is
$ \| A \|_{C^{3}_{\log 3}} = \left( 1^{\log 3} + 2^{\log 3} + 3^{\log 3}\right)^{\frac{1}{\log 3}} \approx 5.48 $
implying that $\| A\|_{C^{n}_{\log n}} > \| A \|$.
Am I doing something wrong without realizing it? I'm confused here. I'd like to show the equivalence of two norms but I'm stuck...