I need help with the following:
Let $A=\left(\begin{array}{cc}a & b \\c & d\end{array}\right)$, with $a\in\mathbb{R}$, $b\in(l^{1})^{*}$, $c\in l^{1}$, and $d\in L(l^{1},l^{1})$.
Let $h\in l^{1}$.
What I'm trying to show is that then
$||Ah||\leq||A||_{1}\cdot||h||$, where $||A||_{1}=\sup_{j}\sum_{i}|A_{ij}|$ (maximum absolute column sum).
My attempt so far: I recalled the definition of the operator norm
$|||A|||=\sup_{h\neq0}\frac{||Ah||}{||h||}$
and looking at itmy inequality seemed to follow by definition, since $||A||_{1}$ is an example of an operator norm, right? Can it be that simple?
Thank you.
Strictly speaking, one has first to show that the right hand side of the defining equation of $\|A\|_1$, which a priori belongs to $\mathbb{R}_{\geq0} \cup \{\infty\}$, is actually a finite number. Then, establishing the inequality $\|Ah\| \leq \|A\|_1\|h\|$ for $h \in \ell^1$, we will deduce that $A \in L(\ell^1,\ell^1)$. Only then can we speak of the operator norm of $A$, and observe that in fact it equals $\|A\|_1$, as we have $|||\,B\,||| = \|B\|_1$ for all $B \in L(\ell^1,\ell^1)$.
Since $(\ell^1)^* \simeq \ell^\infty$, we identify $b \in (\ell^1)^*$ with a bounded sequence $(b_j)_{j=1}^\infty \subset \mathbb{R}$. Similarly, we view $d \in L(\ell^1,\ell^1)$ as an infinite real matrix $(d_{ij})_{i,j=1}^\infty$ for which $\|d\|_1 = \sup_{j\geq1}\sum_{i=1}^\infty|d_{ij}| < \infty$. Finally, we write $c = (c_i)_{i=1}^\infty$ and $h = (h_j)_{j=1}^\infty$. With this notation, we have $$ \|A\|_1 = \sup\left\{|a|+ \sum_{i=1}^\infty|c_i|,\ \sup_{j\geq1}\left(|b_j| + \sum_{i=1}^\infty|d_{ij}|\right)\right\}.$$ Since $|a|+\sum_{i=1}^\infty|c_i| = |a| + \|c\| <\infty$ and $$\sup_{j\geq1}\left(|b_j| + \sum_{i=1}^\infty |d_{ij}|\right) \leq \sup_{j\geq1}|b_j| + \sup_{j\geq1}\sum_{i=1}^\infty |d_{ij}| = \|b\|_\infty + \|d\|_1 < \infty,$$ we have $\|A\|_1 < \infty$ as well. Now, consider $\|Ah\|$. We have $$\begin{multline} \|Ah\| = \left|ah_1 + \sum_{j=1}^\infty b_jh_{j+1}\right| + \sum_{i=1}^\infty\left|c_ih_1 + \sum_{j=1}^\infty d_{ij}h_{j+1}\right| \leq\\ |a|\,|h_1| + \sum_{j=1}^\infty |b_j|\,|h_{j+1}| + \sum_{i=1}^\infty\left(|c_i|\,|h_1|+\sum_{j=1}^\infty|d_{ij}|\,|h_{j+1}|\right). \end{multline}$$ Re-grouping the terms in the above expression and changing the order of summation in the convergent double series with non-negative terms $\sum_{i=1}^\infty\sum_{j=1}^\infty |d_{ij}|\,|h_{j+1}|$, we obtain the estimate $$\begin{multline} \|Ah\| \leq \left(|a| + \sum_{i=1}^\infty|c_i|\right)|h_1| + \sum_{j=1}^\infty\left(|b_j| + \sum_{i=1}^\infty|d_{ij}|\right)|h_{j+1}| \leq\\ \|A\|_1|h_1| + \sum_{j=1}^\infty\|A\|_1|h_{j+1}| = \|A\|_1\|h\|, \end{multline}$$ as required.