Suppose $k \in C\left( \left[ 0,1 \right] \times \left[ 0,1 \right],\mathbb{C} \right)$ is a given continuous complex function on $\left[ 0,1 \right] \times \left[ 0,1 \right]$. Let $B \in B\left( C\left( \left[ 0,1 \right] \right) \right)$ be bounded linear map from continuous complex functions on $\left[ 0,1 \right]$ to the same space, given by
$\left( Bu \right)\left( s \right) = \int_0^s k\left( s,t \right)u\left( t \right) \; dt,$
for $u \in C\left( \left[ 0,1 \right] \right),s \in \left[ 0,1 \right]$. Determine the spectral radius and spectrum of $B$.
My attempt:
$\left\| Bu \right\|_\infty = \left\| \int_0^s k\left( s,t \right)u\left( t \right)dt \right\|_\infty = \max \limits_{s \in \left[ 0,1 \right]} \left| \int_0^s k\left( s,t \right)u\left( t \right) \; dt \right| \leqslant \max_{s \in \left[ 0,1 \right]} \int_0^s \left| k\left( s,t \right)u\left( t \right) \right|dt \leqslant \left\| u \right\|_\infty \max_{s \in \left[ 0,1 \right]} \int_0^s \left| k\left( s,t \right) \right| \; dt \Rightarrow \left\| B \right\|_\infty \leqslant \max_{s \in \left[ 0,1 \right]} \int_0^s \left| k\left( s,t \right) \right|\;dt $
which implies that for spectral radius $\nu \left( B \right)$ of $B$, we have $\nu \left( B \right) \leqslant \max\limits_{s \in \left[ 0,1 \right]} \int_0^s \left| k\left( s,t \right) \right| \; dt $.
However, it seems to me that I am not any closer to the solution.