Measures on spaces of operators and isometries

Let $\mu$ be a finite Borel (in the strong operator topology) measure on the space $B(E, F)$ of bounded linear operator from $E$ into $F$; $E$, $F$ being Banach spaces. Suppose that either $E=C(K)$, $F$ arbitrary, $p>1$ or $E=F=L^q(Y)$, $p>1$, $q>1$, $q\not\in[p,2]$. Suppose next that $\|e\|^p=\int\|Te\|^p\,d\mu(T)$ for every $e\in E$. Then $\mu$ is supported on scalar multiples of isometries.