Reference: Covariance operator is compact

Question

I am looking for a reference for the following theorem:

Let $(X,\Vert \cdot \Vert)$ be a separable Banach space and $C: X^{\ast} \rightarrow X \subseteq X^{\ast \ast}, f \mapsto \int_X f(x) \cdot - ~ d\mu(x) =: q(f, -)$ be the covariance operator of a Gaussian measure on $X$. Then $X$ is compact, where $X^{\ast}$ carries the operator norm topology and $X$ the norm topology.

For finite dimension this is trivial.

For a separable Hilbert spaces one can even show that $C$ is trace class by choosing an orthonormal basis $\{e_n \}_{n \in \mathbb{N}}$ and

$$\int_H \Vert x \Vert_H^2 d \mu(x) = \sum_{n = 1}^{\infty} \int_H \langle x, e_n \rangle^2 d \mu(x) = \sum_{n = 1}^{\infty} q(e_n, e_n) = \sum_{n = 1}^{\infty} \langle C e_n, e_n \rangle = \text{tr} ~ C$$

In his notes on SPDE, Martin Hairer gives the following hint:

Proceed by contradiction by first showing that if $C$ wasn’t compact, then it would be possible to find a constant $c > 0$ and a sequence of elements $(f_n)_{n \in \mathbb{N}}$ such that $\Vert f_n \Vert = 1$, $q(f_k, f_n) = 0$ for any $k \neq n$ and $q(f_n, f_n) \geq c$ for every $n \geq 1$.

Conclude that if this was the case, then the law of large numbers applied to the sequence of random variables $(f_n)_{n \in \mathbb{N}}$ would imply that $\sup_{n \in \mathbb{N}} f_n(x) = \infty$ for $\mu$-almost every $x \in X$, thus obtaining a contradiction with the fact that $\sup_{n \in \mathbb{N}} \vert f_n(x) \vert \leq \Vert x \Vert < \infty$ for $\mu$-almost every $x \in X$.

Answer 1

One approach is to note that $C$ is essentially the embedding of $X^*$ into the Cameron-Martin space $H \subset X$. Now the result follows from the fact that $H$ is compactly embedded into $X$. This can be found as:

• Lemma I.4.5 of Kuo, Hui-Hsiung, Gaussian measures in Banach spaces, Lecture Notes in Mathematics. 463. Berlin-Heidelberg-New York: Springer-Verlag. VI, 224 p. DM 23.00 (1975). ZBL0306.28010.

• Corollary 3.2.4 of Bogachev, Vladimir I., Gaussian measures. Transl. from the Russian by the author, Mathematical Surveys and Monographs. 62. Providence, RI: American Mathematical Society (AMS). xii, 433 p. (1998). ZBL0913.60035.

Another approach, similar to what I did in Proposition 4.16 of these lecture notes, is to first prove that $\int_X \|x\|^2\,d\mu(x) < \infty$, e.g. as a corollary of the much stronger Skorokhod or Fernique theorems. Now suppose we have a sequence $f_n$ in the unit ball of $X^*$. By Alaoglu, we can pass to a subsequence converging weak-* to some $f$. Now since the $f_n$ have norm at most one, then as functions on $X$, they are dominated by the square-integrable function $\|\cdot\|$. Hence by dominated convergence, they converge to $f$ strongly in $L^2(\mu)$. Now Cauchy-Schwarz gives, for any $g \in X^*$ with $\|g\| \le 1$, $$|q(f_n - f, g)| \le \|f_n - f\|_{L^2} \|g\|_{L^2} \le \|f_n - f\|_{L^2} \left(\int \|x\|^2\mu(dx)\right)^{1/2}$$ where the right side approaches 0 uniformly in $g$, so we have $q(f_n - f, \cdot) \to 0$ in $X^{**}$.