Let $x \in \mathbb{R}^\mathbb{\mathbb{N}}$ be a centered random variable, distributed according to a Borel probability measure $\mu$.
Suppose additionally that $x$ has bounded covariance operator, in the sense that for any $y$ with finitely many nonzero entries (i.e., a functional in the topological dual to $\mathbb{R}^\mathbb{N}$), we have $$ \int \Big(\sum_{j=1}^\infty x_j y_j\Big)^2 \, \mu(dx) \leq \sum_{j=1}^\infty y_j^2. $$
Consider a compact operator $T$ on the sequence space $\ell^2$, so that for $z \in \ell^2$, $$ Tz = \sum_{i=1}^\infty \sigma_i \langle z, u_i \rangle_{\ell^2} v_i. $$ where above $\{u_i\}, \{v_i\}$ are two orthonormal sequences in $\ell^2$, and $\sigma_i$ are nonnegative real scalars which tend to $0$ as $i \to \infty$.
I want to understand when it is possible to extend $T$ so that we can have the random variable $Tx$, which we define by $$ Tx:= \sum_i \sigma_i \langle x, u_i \rangle v_i. $$ Is this random variable well defined as a random element of $\mathbb{R}^\mathbb{N}$? (Note, above I am writing $\langle x, u_i \rangle = \sum_j x_j u_{ij}$.)