here is the standard definition of L_2-space of matrix-valued functions: it's those measurable F that satisfy
(or equivalently, , or )
(here is the adjoint of )
clearly this implies
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...