Find larger space such that identity covariance is trace class

Question

Assume I have the identity covariance operator $C = \mathrm{Id}$ on the Sobolev space $H^1([0,1]^2)$. Can I find a larger space $H$ with $H^1([0,1]^2) \subset H$ such that this covariance $C$ is trace class on $H$. Some remarks in Da Prato are suggesting that this holds true.

Answer 1

You can choose any orthonormal basis $e_i$ of $H^1([0,1]^2)$.

You can then define an inner product

$$\langle f, g\rangle_s = \sum_{i=1}^\infty i^{2s} \langle f, e_i \rangle \langle g, e_i \rangle$$

for negative $s$. Define $U^s$ as the closure of $H^1$ with respect to the norm $\|\cdot\|_s$ implied by the scalar product above. On $H^s$, $e_i i^{-s}$ forms an ONB. The trace of $C$ on $H^s$ is

$$tr_s(C) = \sum C(i^{-s}e_i, i^{-s}e_i) = \sum i^{-2s}$$ which is finite if $s > 1/2$, therefore $C$ will have finite trace on all $U^{-s}$ for $s > 1/2$.