L_2 space for matrix-valued functions

hi,

here is the standard definition of L_2-space of matrix-valued functions: it's those measurable F that satisfy

(i) $\displaystyle \int_0^1 ||F(x)||^2dx <\infty$

(or equivalently, $\displaystyle \int_0^1 ||F(x)^*F(x)||dx <\infty$, or $\displaystyle \int_0^1 trace(F(x)^*F(x))dx <\infty$)

(here $\displaystyle F^*$ is the adjoint of $\displaystyle F$)

clearly this implies

(ii) $\displaystyle \left\|\int_0^1 F(x)^* F(x)dx\right\| <\infty$

what about the converse -- does (ii) necessarily implies (i), i.e. is (ii) enough for the function to be in L_2?

for the scalar case (i)=(ii), but I suspect (and afraid) that this fails for matrices...

...help...