I'm reading Stochastic Differential Equations in Infinite Dimensions and don't understand what the authors do in Chapter 2.3.1. Let me introduce the necessary objects: Let
- $K$ and $H$ be real Hilbert spaces
- $Q$ be a bounded, linear, nonnegative and symmetric operator on $K$
- $K_Q:=Q^{1/2}K$
- $\operatorname{HS}(A,B)$ denote the space of Hilbert-Schmidt operators from $A$ to $B$
The authors state that with $\phi\in\operatorname{HS}(K_Q,H)$ and $\psi\in H$, we've got $\phi^\ast\psi\in\operatorname{HS}(K_Q,\mathbb R)$, where $\phi^\ast$ denotes the adjoint of $\phi$.
That doesn't make any sense to me. By definition of the adjoint, $\phi^\ast$ is a bounded, linear operator from $H$ to $K_Q$. So, we should have $\phi^\ast\psi\in K_Q$.
I've noticed that the authors use a different definition of the adjoint which is introduced in Chapter 1.2. With that definition, $\psi^\ast$ is a bounded, linear operator from $H'$ to $K_Q'$, where $A'$ denotes the topological dual space of $A$. [This definition of the adjoint is usually been used for operators between Banach spaces; I don't understand why they use it here].
However, that still doesn't make sense to me. I don't even see that $\phi^\ast\psi$ is a well-defined expression.
So, what do the authors mean?