Spectral characterization of induced operator norm Consider $\mathbb{R}^n$ with the $l^1$ norm and the induced operator norm $\| \cdot \|$ on linear maps $T:  \mathbb{R}^n \rightarrow \mathbb{R}^n$. Can $\|T\|$ be characterized somehow by the spectrum of $T$ (or that of an operator related to $T$)?
 A: By $(e_j)_{j=1}^n$ we denote the standard basis of $\ell_1^n$. By $t_{ij}$ the matrix of $T$ in this basis.
For all $x\in\ell_1^n$ we have
$$
\Vert T(x)\Vert_1
=\sum\limits_{i=1}^n\left|\sum_{j=1}^n t_{ij} x_j\right|
\leq\sum\limits_{i=1}^n\sum_{j=1}^n |t_{ij}| |x_j|
=\sum_{j=1}^n\sum\limits_{i=1}^n |t_{ij}| |x_j|
\leq\sum_{j=1}^n|x_j|\sum\limits_{i=1}^n |t_{ij}| \\
\leq\sup_{j\in\{1,\ldots,n\}}\sum\limits_{i=1}^n |t_{ij}| \sum_{j=1}^n|x_j|
\leq\left(\sup_{j\in\{1,\ldots,n\}}\sum\limits_{i=1}^n |t_{ij}|\right) \Vert x\Vert_1
$$
Hence
$$
\Vert T\Vert\leq \sup_{j\in\{1,\ldots,n\}}\sum\limits_{i=1}^n |t_{ij}|\tag{2}
$$
On the other hand $\Vert e_j\Vert_1=1$ and $\Vert T(e_j)\Vert_1=\sum\nolimits_{i=1}^n|t_{ij}|$, so for all $j\in\{1,\ldots,n\}$. So we get
$$
\Vert T\Vert
\geq\sup_{j\in\{,1,\ldots,n\}}\frac{\Vert T(e_j)\Vert_1}{\Vert e_j\Vert_1}
=\sup_{j\in\{,1,\ldots,n\}}\sum\limits_{i=1}^n|t_{ij}|\tag{2}
$$
From  $(1)$ and $(2)$ we get
$$
\Vert T\Vert=\sup_{j\in\{,1,\ldots,n\}}\sum\limits_{i=1}^n|t_{ij}|
$$
