# Convergence of random operators

I'm a statistician not versed in functional analysis and operator theory. I wish that I might not find a wrong place for my question. All my questions are trivial in the scalar time series case, but the difficulty arises due to the infinite-dimensional functional(operator) setting. Any tiny or partial comment and insight would be enormously appreciated.

I consider a functional time series $$(f_t)_{t=1}^{T}$$ taking values in an infinite-dimensional separable Hilbert space $$H$$, specified by $$f_t=Af_{t-1}+\varepsilon_t$$, where $$A$$ is a Hilbert-Schmidt operator whose operator norm $$\Vert A\Vert_{ \infty}$$ is less than 1 so that $$(f_t)_{t=1}^{T}$$ is stationary.

Then, I think of the variance operator $$\Sigma_f = \mathbb{E}(f_t\otimes f_t)$$ of the time series, where $$\otimes$$ denotes the tensor product in $$H$$. Then, it is well-known that $$\Sigma_f$$ is a self-adjoint and trace-class operator, and it admits the spectral representation: $$\Sigma_f = \sum_{i=1}^{\infty}\lambda_i(v_i\otimes v_i)$$, where $$(\lambda_i,v_i)_{i=1}^{\infty}$$ is the eigenvalue and (orthonormalized) eigenfunction pairs of $$\Sigma_f$$, and $$\lambda_i \downarrow 0$$ as $$i \rightarrow \infty$$.

Thus, $$\Sigma_f^{-1}$$ is an unbounded operator, but it seems to be usually assumed in the literature that $$\Vert \Sigma_f^{-\frac{1}{2}}A \Vert_{\infty}<\infty$$.(If you should be interested, see this paper.)

Then, I am given consistent estimators $$\hat{A}$$ and $$\hat\Sigma_f$$ respectively for $$A$$ and $$\Sigma_f$$. Here, $$\hat A$$ is of finite rank for each $$T$$, and $$\Vert \hat A -A\Vert_{\infty} \rightarrow 0$$ in probability as $$T \rightarrow \infty$$. Also, $$\hat \Sigma_f$$ is self-adjoint and of finite rank for each $$T$$, and $$\Vert\hat \Sigma_f - \Sigma_f \Vert_{\infty} \rightarrow 0$$ in probability as $$T \rightarrow \infty$$.

Since both $$\hat A$$ and $$\hat \Sigma_f$$ are of finite rank for each $$T$$, it always holds true that $$\Vert \hat\Sigma_f^{-\frac{1}{2}}\hat A \Vert_{\infty} < \infty$$ for each T.

Then, my three questions are

Q1: Is it guaranteed that $$\Vert \hat\Sigma_f^{-\frac{1}{2}}\hat A - \Sigma_f^{-\frac{1}{2}}A \Vert_{\infty} \rightarrow 0$$ in probability as $$T\rightarrow \infty$$ ? If not, could we have at least that $$\Vert \hat\Sigma_f^{-\frac{1}{2}}\hat A \Vert_{\infty} = O_p(1)$$ as $$T \rightarrow \infty$$ ?

Q2: If another estimator $$\tilde \Sigma_f$$ for $$\Sigma_f$$ is such that $$\Vert \Sigma_f - \tilde \Sigma_f + A\Sigma_f A^{*} - \hat A\tilde \Sigma_f \hat A^{*} \Vert_{\infty} \rightarrow 0$$ in probability as $$T \rightarrow \infty$$, where $$A^{*}$$ and $$\hat A^{*}$$ are respectively the adjoints of $$A$$ and $$\hat A$$, then does it imply that $$\Vert \Sigma_f - \tilde \Sigma_f \Vert_{\infty} \rightarrow 0$$ in probability as $$T \rightarrow \infty$$ ?

Q3: Could you please let me know, if any, book, monograph, or paper dealing with such materials in a relatively easily accessible way ?

Any tiny or partial comment and insight would be enormously appreciated!