Trace term in the Itō formula

Question

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
• $Q\in\mathfrak L(K)$ be nonnegative and symmetric
• $K_Q:=Q^{1/2}K$
• $(\Omega,\mathcal A,\operatorname P)$ be a probability space
• $\Phi:\Omega\times[0,\infty)\to\operatorname{HS}(K_Q,H)$ and $\varphi:\Omega\times[0,\infty)\to H$
• $F:[0,\infty)\times H\to\mathbb R$ and $F_{xx}$ be the second Fréchet derivative of $F$ with respect to the second variable

I don't understand the term $$\operatorname{tr}\left[F_{xx}(t,x)\left(\Phi_tQ^{\frac 12}\right)\left(\Phi_tQ^{\frac 12}\right)^\ast\right]\tag 1$$ which occurs in equation (2.53).

By definition, $F_{xx}$ is an element of $\mathfrak L(H,\mathfrak L(H,\mathbb R))$. However, the authors obviously use the fact that $\mathfrak L(H,\mathbb R)\cong H$ and hence $F_{xx}$ can be identified with an element of $\mathfrak L(H)$. With this interpretation, we've got $$\underbrace{F_{xx}(t,x)}_{\in\mathfrak L(H)}\underbrace{\underbrace{\left(\Phi_tQ^{\frac 12}\right)}_{\in\mathfrak L(K_Q,H)}\underbrace{\left(\Phi_tQ^{\frac 12}\right)^\ast}_{\in\mathfrak L(H,K_Q)}}_{\in\mathfrak L(H)}\in\mathfrak L(H)\;.$$ Thus, at least it makes sense to talk about the trace of this expression. However, can we rewrite the expression $(1)$ without the identification?

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Above $\mathfrak L(A,B)$ and $\operatorname{HS}(A,B)$ denote the space of bounded, linear operators and Hilbert-Schmidt operators from $A$ to $B$, respectively. Moreover, $\mathfrak L(A):=\mathfrak L(A,A)$ and $L^\ast$ denotes the adjoint of a bounded, linear operator $L$.

Answer 1

The answer is: Yes, we can.

It's not important that we talk about real Hilbert spaces. So, let's replace $\mathbb R$ by $\mathbb K\in\left\{\mathbb C,\mathbb R\right\}$ in the question.

Let

• $(t,x)\in[0,\infty)\times H$
• $(e_n)_{n\in\mathbb N}$ and $(f_n)_{n\in\mathbb N}$ be orthonormal bases of $K$ and $H$, respectively

Note that $$\tilde\Phi:=\Phi_tQ^\frac12\in\mathfrak L(K,H)$$ is trace class and that ${\rm D}^2F(t,x):=F_{xx}(t,x)\in\mathfrak L(H,\mathfrak L(H,\mathbb K))$ can be identified with $L\in\mathfrak L(H)$, where $$Lu:=\sum_{n\in\mathbb N}\overline{{\rm D}^2F(t,x)uf_n}f_n\;\;\;\text{for }u\in H\;.$$ It's easy to check that $$\langle v,Lu\rangle={\rm D}^2F(t,x)uv\;\;\;\text{for all }u,v\in H\;.$$ Thus, we can conclude that $$\operatorname{tr}L\tilde\Phi\tilde\Phi^\ast=\sum_{n\in\mathbb N}\langle\tilde\Phi e_n,L\tilde\Phi e_n\rangle_H=\sum_{n\in\mathbb N}{\rm D}^2F(t,x)\left(\tilde\Phi e_n\right)\left(\tilde\Phi e_n\right)\;.$$