Understanding Operator Norm of Matrices Let $X$ denote the vector space of $n\times n$ complex matrices. To every matrix $A\in X$ one can associate two operator norms:


*

*Thinking of $A$ as a map $A\colon \mathbb{C}^n\to \mathbb{C}^n$ or $A\in L(\mathbb{C}^n)$ in short, we define $\|A\|_{L(\mathbb{C}^n)}=\sup_{v\in \mathbb{C}^n, \|v\|=1}{\|Av\|}$.

*Thinking of $A$ as a map $A\colon X\to X$ or $A\in L(X)$ in short, we define $\|A\|_{L(X)}=\sup_{B\in X, \|B\|_{L(\mathbb{C}^n)}=1}{\|AB\|}$.


It is well-known that $\|A\|_{L(\mathbb{C}^n)}=s_1(A)$ where $s_1$ is the largest singular value of $A$. My questions are:


*

*Is there is a (spectral?) characterization of $\|A\|_{L(X)}$? or perhaps a relation between the two above norms?

*Is there any application of the second norm (in, say functional analysis)? I will explain below why I started thinking about this norm in the first place.

*I was trying to understand the following paragraph that lead to my questions above. "If $X$ is equipped with the operator norm $\|\cdot\|_{L(\mathbb{C}^n)}$, then its dual space $X^*$ is the space $X$ equipped with the trace norm $\|\cdot\|_1=\text{ sum of singular values}$, and vice versa."


I still don't see why (3) has to be true, but if necessary, and if it turns out to be irrelevant to the other two questions, I can ask it in a separate question.
Any help is appreciated.
 A: I will assume that the norm $\|AB\|$ in the definition of $\|A\|_{L(X)}$ is $\|AB\|_{L(\mathbb C^n)}$. 
The two norms are equal: you have, by definition, 
$$
\|A\|_{L(X)}\leq \|A\|_{L(\mathbb C^n)},
$$
since $\|AB\|_{L(\mathbb C^n)}\leq \|A\|_{L(\mathbb C^n)}\,\|B\|_{L(\mathbb C^n)}=\|A\|_{L(\mathbb C^n)}$.
Conversely,
$$
\|A\|_{L(X)}\geq\left\|A\,\frac{A^*}{\|A^*\|}\right\|_{L(\mathbb C^n)}
=\frac{\|AA^*\|_{L(\mathbb C^n)}}{\|A\|_{L(\mathbb C^n)}}
=\frac{\|A\|^2_{L(\mathbb C^n)}}{\|A\|_{L(\mathbb C^n)}}
=\|A\|_{L(\mathbb C^n)}.
$$
Regarding the dual $X^*$, it is not hard to see that one can write 
$$
X^*=\{\text{Tr}(B\,\cdot):\ B\in X\},
$$
giving the identification of every functional in $X^*$ with a matrix $B$. Now let us write $f_B=\text{Tr}(B\cdot)$ and let us look at the norm:
$$
\|f_B\|=\sup\{|f_B(A)|:\ \|A\|_{L(\mathbb C^n)}=1\}
=\sup\{|\text{Tr}(BA)|:\ \|A\|_{L(\mathbb C^n)}=1\}
=\|B\|_1.
$$
Here is a proof of the last equality: write $B=UDV$ the singular value decomposition. Using that unitaries preserve the norm,
$$
\sup\{|\text{Tr}(BA)|:\ \|A\|_{L(\mathbb C^n)}=1\}
=\sup\{|\text{Tr}(DA)|:\ \|A\|_{L(\mathbb C^n)}\}
=\sup\{\left|\sum_jD_{jj}A_{jj}\right|:\ \|A\|_{L(\mathbb C^n)}=1\}=
\sum_jD_{jj}=\|B\|_1.
$$
A: Expansion on the existing answer
by taking a more general view on $X=L(\mathbb{C}^n)$:
When equipped with the operator norm, then $X$ is a $C^*$-algebra  (with Matrix multiplication as product, and Transpose + Complex conjugate as involution, just to make sure), and it's reasonable that Martin Argerami added the corresponding tag.
He shows the equality
$$\|A\|_{L(X)}:=\sup_{B\in X,\,\|B\|_X=1}\|AB\|_X\;\overset{!}{=}\;\|A\|_X$$
of the two norm expressions, and this equality holds true for every  $\,C^*$-Algebra. With the very same proof: An inspection shows that only $\,C^*$-norm features are used, but no particular property depending on $L(\mathbb{C}^n)$.
W.r.to applications of this fact:
It is an ingredient of "Unitisation of a $C^*$-Algebra $A$", you may consult [Pedersen: "Analysis now", 4.3.9 Lemma], and here's a sketch of the recipe:
With $A_1$ denoting $A$'s algebraic unitisation one starts with the algebra homomorphism
$$A_1\to L(A),\;(a,\lambda)\longmapsto L_a +\lambda\cdot\operatorname{id}_A$$
in order to embed $A_1$ into $L(A)$, in which a norm and the unit are naturally available.
$L_a$ denotes left multiplication by $a\in A\,$ (and the above homomorphism is injective iff $A$ has no unit). Then the norm is pulled back to $A_1$ where it does the $C^*$-job. The corresponding proof depends on the norm equality under consideration.
You may switch to this
"Unitization" post,
astonishingly having found his home @MO.
IIRC, the fact is also needed in the construction of the Multiplier algebra of a $C^*$-Algebra (which belongs to the unitisation context as well).
