# Schur’s condition for a matrix to be a bounded operator on $l^{2}$

###### Theorem 0.1

Let $B$ be a matrix defined on $T\times T$ for some countable set $T$. If there exists a positive number $C$ such that

 $\sum_{t\in T}|b(s,t)|

then $B$ is a bounded operator on $l^{2}(T)$ with its operator norm $\left\|B\right\|$ less than or equal to $C$.

• Proof.

Let $x$ be a sequence in $l^{2}(T)$. We have

 $\displaystyle\left\|Bx\right\|_{2}^{2}$ $\displaystyle=\sum_{s\in T}\left|\sum_{t\in T}b(s,t)x(t)\right|^{2}$ $\displaystyle\leq\sum_{s\in T}\left[\sum_{t\in T}\sqrt{|b(s,t)|}\left(\sqrt{|b% (s,t)|}|x(t)|\right)\right]^{2}$ $\displaystyle\leq\sum_{s\in T}\left[\left(\sum_{t\in T}|b(s,t)|\right)\left(% \sum_{t\in T}|b(s,t)||x(t)|^{2}\right)\right]$ $\displaystyle\leq C\sum_{s\in T}\sum_{t\in T}|b(s,t)||x(t)|^{2}$ $\displaystyle\leq C\sum_{t\in T}|x(t)|^{2}\sum_{s\in T}|b(s,t)|$ $\displaystyle\leq C^{2}\sum_{t\in T}|x(t)|^{2}.$

Therefore we have $\left\|Bx\right\|_{2}\leq C\left\|x\right\|_{2}$ for all $x\in l^{2}(T)$, hence $\left\|B\right\|\leq C$. $\square$

Title Schur’s condition for a matrix to be a bounded operator on $l^{2}$ SchursConditionForAMatrixToBeABoundedOperatorOnL2 2013-03-22 15:57:18 2013-03-22 15:57:18 Gorkem (3644) Gorkem (3644) 5 Gorkem (3644) Theorem msc 46C05 Schur’s Lemma Schur’s Lemma for infinite matrices