If $k:[a,b]\times[a,b]\rightarrow \mathbb{R}$ is in $L^2([a,b]\times[a,b])$, we can show that the linear operator:
$T_k: L[a,b]\rightarrow L[a,b]$, given by $T_k(f)(s)=\int_{[a,b]}k(s,t)f(t)d\lambda(t)$ is a bounded linear operator. This follows from use of holders inequality. The same proof gives us that $\|T_k\|\le \|k\|_2$. But is there a way to show what the norm actually is? Is the norm equal $\|k\|_2$? Will it be easier to find the norm if we assume that k is continous on $[a,b]\times [a,b]$?
I tried something but couldn't finish it: We have that :
$\|T_k(f)\|_2^2=\int_{[a,b]}|\int_{[a,b]}k(s,t)f(t)d\lambda(t)|^2d\lambda(s)$. To show boundedness then you would go from here and use monotonicity for integrals and Holder, but in order to find the operator norm, we have to choose a smart f where $\|f\|_2\le 1$, but $\|T_k(f)\|_2^2$is close to for instance $\|k\|_2^2$(if this indeed turns out to be the operator norm?).
Do you guys have any tips? PS: If you know what to do in the case where k is continuous I am also interested in that.