Random operators

Question

Let $(\Omega, \mathcal F,P)$ be a probability spaces and $H$ be a Hilbert space. By a random operator $A$ from $H$ to $H$ we mean a linear continuous mapping from $H$ into the Frechet space $L_0^H (\Omega, \mathcal F,P)$ of all $H$-valued random variables. My question is: How Can we define $|A|$ and the adjoint $A^*$?

Answer 1

We can view the random operator as a map from $H$ to the space of random variables, $$ A \colon H\rightarrow (\Omega\rightarrow H) $$ but also as an operator-valued random variable, $$ X_A \colon \Omega\rightarrow (H\rightarrow H) $$ by letting $$ X_A(\omega)(x)=A(x)(\omega). $$ Thus, given $\omega\in\Omega$, $X_A(\omega)$ is an operator on $H$ and hence, I guess, has an adjoint $(X_A(\omega))^*$ and an absolute value $|X_A(\omega)|$. Then you could also say that $$ A^*(x)(\omega) = (X_A(\omega))^*$$ and $$ |A|(x)(\omega) = |(X_A(\omega)|$$