---
author:
- |
    Houman Owhadi and Clint Scovel\
    California Institute of Technology
bibliography:
- 'refs.bib'
title: Conditioning Gaussian measure on Hilbert space
---

Introduction {#sec_intro}
============

For a Gaussian measure $\mu$ with injective covariance operator $C$ on a direct sum of finite dimensional Hilbert spaces $H=H_{1}\oplus H_{2}$, the conditional measure associated with conditioning on the value of $H_{2}$ can be computed in terms of the Schur complement corresponding to the partitioning of the covariance matrix $C$, see Cottle [@cottle1974manifestations] for a review. Evidently, the natural extension to infinite dimensions of the Schur complement is the [*shorted operator*]{}, first discovered by Krein [@krein1947theory] and developed in Anderson and Trapp [@anderson1975shorted] based on results on operator ranges of Douglas [@douglas1966majorization] and Fillmore and Williams [@fillmore1971operator]. For related results, and a history, see Pekarev [@pekarev1992shorts]. However, the connection between the shorted operator and the covariance operator of the conditional Gaussian measure on an infinite dimensional Hilbert space appears yet to be established. Indeed, Hairer, Stuart, Voss, and Wiber [@hairer2005analysis Lem. 4.3], see also Stuart [@Stuart:2010 Thm. 6.20], characterizes the conditional measure through a measurable extension result of Dalecky and Fomin [@dalet︠s︡kiui1991measures Thm. II.3.3] of an operator defined on the Cameron-Martin reproducing kernel Hilbert space. For other representations, see Mandelbaum [@mandelbaum1984linear], LaGatta [@lagatta2013continuous], and Tarieladze and Vakhania’s [@tarieladze2007disintegration] extension of the optimal linear approximation results of Lee and Wasilkowski [@lee1986approximation] from finite to infinite rank. Tarieladze [@tarieladzeinformation] asserts that this latter result extends one in the Information-Based Complexity of Traub, Wasilkowski and Wozniakowski [@traubniakowski] which is relevant to Grid Computing as described in Foster and Kesselman [@foster2003grid]. The primary purpose of this paper is to instead represent the conditional measure in terms of the shorted operator. We provide two distinct proofs of this representation. The first uses the theory of Gaussian Hilbert spaces and a characterization of the shorted operator by Andersen and Trapp. The second proof, corresponding to the secondary purpose of this paper, uses recent developments by Corach, Maestripieri and Stojanoff on the relationship between the shorted operator and $A$-symmetric oblique projections. This latter approach has the advantage that it facilitates a general approximation technique that not only can be used to approximate the covariance operator but the conditional expectation operator. This is accomplished through the development of an approximation theory for the shorted operator in terms of oblique projections followed by an application of the martingale convergence theorem. Although the proofs are not fundamentally difficult, the result (which appears to have been missed in the literature) provides a simple characterization of the conditional measure, leading to significant approximation results. For instance, the attainment of the main result through the martingale approach feeds back a strengthening of the approximation theorem for the shorted operator that was developed for that purpose: when the operator is trace class the approximation improves from weak convergence to convergence in trace norm.

Let us review the basic results on Gaussian measures on Hilbert space. A measure $\mu$ on a Hilbert space $H$ is said to be Gaussian if, for each $h \in H$ considered as a continuous linear function $h:H \rightarrow {\mathbb{R}}$ by $h(x):=\langle h,x\rangle, x \in H$, we have that the pushforward measure $h_{*}\mu$ is Gaussian, where we say that a Dirac measure is Gaussian. For a Gaussian measure $\mu$, its mean $m$ is defined by $$\langle h ,m\rangle :=\int_{H}{\langle h, x\rangle d\mu(x)},  \quad  h \in H \,$$ and its covariance operator $C:H \rightarrow  H $ is defined by $$\langle Ch_{1},h_{2}\rangle :=\int_{H}{\langle h_{1}, x\rangle \langle h_{2}, x\rangle d\mu(x)}-
\langle  h_{1}, m\rangle \langle  h_{2}, m\rangle ,  \quad h_{1}, h_{2} \in H \, .$$ A Gaussian measure has a well defined mean and a continuous covariance operator, see e.g. Da Prato and Zabczyk [@da2014stochastic Lem. 2.14]. Finally, Mourier’s Theorem [@mourier1953elements], see Vakhania, Tarieladze and Chobanyan [@vakhania1987probability Thm. IV.2.4], asserts, for any $m \in H$ and any positive symmetric trace class operator $C$, that there exists a Gaussian measure with mean $m$ and covariance operator $C$, and that all Gaussian measures have a well defined mean and positive trace class covariance operator. This characterization also follows from Sazonov’s Theorem [@sazonov1958remark Thm. 1].

Since separable Hilbert spaces are Polish, it follows from the product space version, see e.g. Dudley [@Dudley:2002 Thm. 10.2.2], of the theorem on the existence and uniqueness of regular conditional probabilities on Polish spaces, that any Gaussian measure $\mu$ on a direct sum $H=H_{1}\oplus H_{2}$ of separable Hilbert spaces has a regular conditional probability, that is there is a family $\mu_{t}, t \in H_{2}$ of conditional measures corresponding to conditioning on $H_{2}$. Moreover, Tarieladze and Vakhania [@tarieladze2007disintegration Thm. 3.11] demonstrate that the corresponding family of conditional measures are Gaussian. Bogachev’s [@bogachev1998gaussian Thm. 3.10.1] theorem of normal correlation of Hilbert space valued Gaussian random variables shows that if two Gaussian random vectors $\xi$ and $\eta$ on a separable Hilbert space $H$ are jointly Gaussian in the product space, then ${\mathbb{E}}[\xi|\eta]$ is a Gaussian random vector and $\xi={\mathbb{E}}[\xi|\eta]+\zeta$ where $\zeta$ is Gaussian random vector which is independent of $\eta$. Consequently, for any two vectors $h_{1}, h_{2} \in H$ we have $$\begin{aligned}
{\mathbb{E}}\Bigl[\bigl\langle \xi-{\mathbb{E}}[\xi|\eta], h_{1}\bigr\rangle\bigl\langle \xi-{\mathbb{E}}[\xi|\eta], h_{2}\bigr\rangle\big|
\eta\Bigr]&=&{\mathbb{E}}\Bigl[\bigl\langle\zeta , h_{1}\bigr\rangle\bigl\langle \zeta, h_{2}\bigr\rangle\big|
\eta\Bigr]\\
&=&{\mathbb{E}}\Bigl[\bigl\langle\zeta, h_{1}\bigr\rangle\bigl\langle \zeta, h_{2}\bigr\rangle\Bigr]\end{aligned}$$ and so we conclude that, just as in the finite dimensional case, the conditional covariance operators are independent of the values of the conditioning variables.

Since both proof techniques will utilize the characterization of conditional expectation as orthogonal projection, we introduce these notions now. Consider the Lebesgue-Bochner space $L^{2}(H,\mu,\mathcal{B}(H))$ space of (equivalence classes) of $H$-valued Borel measurable functions on $H$ whose squared norm $${\Vert f \Vert}^{2}_{L^{2}(H,\mu,\mathcal{B}(H))}:=\int_{H}{{\Vert f(x) \Vert}^{2}_{H}d\mu(x)}$$ is integrable. For a sub ${\sigma}$-algebra $\Sigma \subset \mathcal{B}(H)$ of the Borel ${\sigma}$-algebra, consider the corresponding Lebesgue-Bochner space $L^{2}(H,\mu,\Sigma)$. As in the scalar case, one can show that $L^{2}(H,\mu,\mathcal{B}(H))$ and $L^{2}(H,\mu,\Sigma)$ are Hilbert spaces and that $L^{2}(H,\mu,\Sigma) \subset L^{2}(H,\mu,\mathcal{B}(H))$ is a closed subspace. Then, if we note that contractive projections on Hilbert space are orthogonal, see Rao [@rao2011foundations Rmk. 9, pg. 51], it follows from Sundaresan [@Sundaresan Prop. 4], see Diestel and Uhl [@DiestelUhl Thm. V.1.4], that conditional expectation amounts to orthogonal projection.

Shorted Operators
=================

A symmetric operator $A:H \rightarrow H$ is called [*positive*]{} if $\langle Ax, x\rangle \geq 0$ for all $x \in H$. We denote by $L_{+}(H)$ the set of positive operators and we denote such positivity by $A \geqq 0$. Positivity induces the (Löwner) partial order $\geqq $ on $L_{+}(H)$. For a closed subspace $S \subset H$ and a positive operator $A\in L_{+}(H)$ consider the set $$\mathcal{H}(A,S):=\bigl\{X \in L_{+}(H):  X \leqq A\,\,  \text{and} \,\,  R(X)\subset S \bigr\}\, .$$ Then, according to Pekarev [@pekarev1992shorts], Krein [@krein1947theory] and later Anderson and Trapp [@anderson1975shorted] showed that $\mathcal{H}(A,S)$ contains a maximal element, which we denote by $\mathcal{S}(A)$ and call the [*short*]{} of $A$ to $S$. For another closed subspace $T\subset H$, we denote the short of $A$ to $T$ by $\mathcal{T}(A)$. In the proof, Anderson and Trapp [@anderson1975shorted] demonstrate that when $A$ is invertible, that in terms of its $(S,S^{\perp})$ partition representation $$A= \begin{pmatrix}
  A_{SS} &  A_{SS^{\perp}}  \\
  A_{S^{\perp}S} & A_{S^{\perp}S^{\perp}}
 \end{pmatrix}$$ that $A_{S^{\perp}S^{\perp}}$ is invertible and $$\mathcal{S}(A)= \begin{pmatrix}
A_{SS}-A_{SS^{\perp}}A_{S^{\perp}S^{\perp}}^{-1} A_{S^{\perp}S}  &  0  \\
  0 & 0
 \end{pmatrix}\, .$$ It is easy to show that the assertion holds under the weaker assumption that $A_{S^{\perp}S^{\perp}}$ be invertible. Moreover, Anderson and Trapp [@anderson1975shorted Cor. 1] asserts for $A, B \in L_{+}(H)$, that $$A \leqq B \implies \mathcal{S}(A) \leqq\mathcal{S}(B) \, ,$$ that is, $\mathcal{S}$ is monotone in the Löwner ordering. In addition, [@anderson1975shorted Cor. 5] asserts that for two closed subspaces $S$ and $T$, we have $$\mathcal{(S\cap T)}(A) =\mathcal{S}\bigl(\mathcal{T}(A)\bigr)\, .$$ Finally, [@anderson1975shorted Thm. 6] asserts that if $A:H \rightarrow H$ is a positive operator and $S \subset H$ is a closed linear subspace, then $$\label{id_short}
  \bigl\langle \mathcal{S}(A)s,s\bigr\rangle =\inf\Bigl\{\Bigl\langle A
\begin{pmatrix}
  s\\
t
 \end{pmatrix}, \begin{pmatrix}
  s\\
t
 \end{pmatrix} \Bigr\rangle, \, t \in S^{\perp} \Bigr\},\quad \forall s \in S\, .$$ In Section \[sec\_proof1\] we demonstrate how the characterization of the shorted operator combined with the theory of Gaussian Hilbert spaces provides a natural proof of our main result, the following theorem. Here we consider direct sum split $H=H_{1}\oplus H_{2}$, and let $S=H_{1}$ and $S^{\perp}=H_{2}$, so that the short $\mathcal{S}(A)$ of an operator to the subspace $S=H_{1}$ will be written as $\mathcal{H}_{1}(A)$.

\[thm\_main0\] Consider a Gaussian measure $\mu$ on an orthogonal direct sum $H=H_{1}\oplus H_{2}$ of separable Hilbert spaces with mean $m$ and covariance operator $C$. Then for all $t \in H_{2}$, the conditional measure $\mu_{t}$ is a Gaussian measure with covariance operator $\mathcal{H}_{1}(C).$

Oblique Projections
===================

In this section, we will prepare for an alternative proof of Theorem \[thm\_main0\] using oblique projections along with the development of approximations of the covariance operator and the conditional expectation operator generated by natural sequences of oblique projections. To that end, let us introduce some notations. For a separable Hilbert space $H$, we denote the usual, or strong, convergence of sequences by $h_{n} \rightarrow h$ and the weak convergence by $h_{n} \xrightarrow{\omega} h$. Let $L(H)$ denote the Banach algebra of bounded linear operators on $H$. For an operator $A \in L(H)$, we let $R(A)$ denote its range and $ker(A)$ denote its nullspace. Recall the uniform operator topology on $L(H)$ defined by the metric ${\Vert A \Vert}:=\sup_{{\Vert h \Vert}\leq 1}{{\Vert Ah \Vert}}.$ We say that a sequence of operators $A_{n} \in L(H)$ converges strongly to $A \in L(H)$, that is $$A= s\mbox{-}\\lim_{n\rightarrow\infty}{A^{n}}$$ if $A_{n}h \rightarrow Ah$ for all $h \in H$, and we say that $A_{n}\rightarrow A$ weakly or $$A= \omega\mbox{-}\\lim_{n\rightarrow\infty}{A^{n}}$$ if $A_{n}h \xrightarrow{\omega} Ah$ for all $h \in H$. Recall that an operator $A \in L(H)$ is called [*trace class*]{} if the [*trace norm*]{} $${\Vert A \Vert}_{1}:=\sum_{i=1}^{\infty}{\langle |A|e_{i},e_{i}\rangle }\,$$ is finite for some orthonormal basis, where $|A|:=\sqrt{A^{*}A}$ is the absolute value. When it is finite, then $tr(A):=\sum_{i=1}^{\infty}{\langle Ae_{i},e_{i}\rangle} $ is well defined, and for all positive trace class operators $A$ we have $tr(A)={\Vert A \Vert}_{1}$. The trace norm ${\Vert \cdot \Vert}_{1}$ makes the subspace $L_{1}(H) \subset L(H)$ of trace class operators into a Banach space. It is well known that the sequence of operator topologies $$\text{weak} \rightarrow \text{strong} \rightarrow \text{ uniform operator} \rightarrow \text{trace norm}$$ increases from left to right in strength.

For a positive operator $A:H \rightarrow H$, let us define the set of ($A$-symmetric) oblique projections $$\mathcal{P}(A,S^{\perp}):=\bigl\{ Q \in L(H): Q^{2}=Q, \, R(Q)=S^{\perp},\,\, AQ=Q^{*}A\bigr \}$$ onto $S^{\perp}$, where $Q^{*}$ is the adjoint of $Q$ with respect to the scalar product $\langle  \cdot, \cdot\rangle$ on $H$. The pair $(A,S^{\perp})$ is said to be [*compatible*]{}, or $S^{\perp}$ is said to be [*compatible with*]{} $A$, if $\mathcal{P}(A,S^{\perp})$ is nonempty. For any oblique projection $Q \in \mathcal{P}(A,S^{\perp})$, Corach, Maestripieri and Stojanoff [@corach2001schur Prop. 4.2] asserts that for $E:=1-Q$, we have $$\label{e1}
\mathcal{S}(A)=AE=E^{*}AE\, .$$ Moreover, when $(A,S^{\perp})$ is compatible, according to Corach, Maestripieri and Stojanoff [@corach2001schur Def. 3.4], there is a special element $Q_{A,S^{\perp}} \in  \mathcal{P}(A,S^{\perp})$ defined in the following way: by [@corach2001schur Prop. 3.3] and the factorization theorem [@corach2001schur Thm. 2.2] of Douglas [@douglas1966majorization] and Fillmore and Williams [@fillmore1971operator], there is a unique operator $\hat{Q}:S \rightarrow S^{\perp}$ which satisfies $A_{S^{\perp}S^{\perp}}\hat{Q}=A_{S^{\perp}S}$ such that $ker(\hat{Q})=ker(A_{S^{\perp}S})$ and $R(\hat{Q})\subset {\mkern 1.5mu\overline{\mkern-1.5muR(A_{S^{\perp}S^{\perp}})\mkern-1.5mu}\mkern 1.5mu}$. Defining $$\label{id_special}
Q_{A,S^{\perp}}= \begin{pmatrix}
  0& 0\\
\hat{Q}& 1
 \end{pmatrix}\, ,$$ [@corach2001schur Thm. 3.5] asserts that $Q_{A,S^{\perp}}  \in \mathcal{P}(A,S^{\perp})$.

When the pair $(A,S^{\perp})$ is not compatible, we seek an approximating sequence $A^{n}$ to $A$ which is compatible with $S^{\perp}$, such that the limit of $\mathcal{S}(A^{n})$ is $\mathcal{S}(A)$. Although Anderson and Trapp [@anderson1975shorted Cor. 2] show that if $A^{n}$ is a monotone decreasing sequence of positive operators which converge strongly to $A$, that the decreasing sequence of positive operators $\mathcal{S}(A^{n})$ strongly converges to $\mathcal{S}(A)$, the approximation from above by $A^{n}:=A+\frac{1}{n}I$ determines operators which are not trace class, so is not useful for the approximation problem for the covariance operators for Gaussian measures. Since the trace class operators are well approximated from below by finite rank operators one might hope to approximate $A$ by an increasing sequence of finite rank operators. However, it is easy to see that, in general, the same convergence result does not hold for increasing sequences. The following theorem demonstrates, for any positive operator $A$, how to produce a sequence of positive operators $A^{n}$ which are compatible with $S^{\perp}$ such that $\mathcal{S}(A^{n})$ weakly converges to $\mathcal{S}(A)$.

Henceforth we consider a direct sum split $H=H_{1}\oplus H_{2}$, and let $S=H_{1}$ and $S^{\perp}=H_{2}$, so that the short $\mathcal{S}(A)$ of an operator to the subspace $S=H_{1}$ will be written as $\mathcal{H}_{1}(A)$. Let us also denote by $P_{i}:H\rightarrow H$ the orthogonal projections onto $H_{i}$, for $i=1,2$, and let $\Pi_{i}:H\rightarrow H_{i}$ denote the corresponding projections and $\Pi^{*}_{i}:H_{i} \rightarrow H$ the corresponding injections. For any operator $A:H\rightarrow H$, consider the decomposition $$A= \begin{pmatrix}
  A_{11}& A_{12}\\
A_{21}& A_{22}
 \end{pmatrix}$$ where the components are defined by $A_{ij}:=\Pi_{i}A\Pi^{*}_{j},\,  i,j=1,2$.

\[thm\_short\] Consider a positive operator $A:H \rightarrow H$ on a separable Hilbert space. Then for any orthogonal split $H=H_{1}\oplus H_{2}$, and any ordered orthonormal basis of $H_{2}$, we let $H^{n}_{2}$ denote the span of the first $n$ basis elements and let $P^{n}:=P_{H_{1}}+P_{H^{n}_{2}}$ denote the orthogonal projection onto $H_{1}\oplus H^{n}_{2}$. Then the sequence of positive operators $$A^{n}:=P^{n}AP^{n},\quad n=1, \ldots$$ is compatible with $H_{2}$ and $$\mathcal{H}_{1}(A)= \omega\mbox{-}\\lim_{n\rightarrow\infty}{ \mathcal{H}_{1}(A^{n})}\, .$$

For an increasing sequence $A^{n}$ of positive operators converging strongly to $A$, the monotonicity of the shorting operation implies that the sequence $\mathcal{H}_{1}(A^{n})$ is increasing, and therefore Vigier’s Theorem, see e.g. Halmos [@halmos1982hilbert Prb. 120], implies that the sequence $\mathcal{H}_{1}(A^{n})$ converges strongly. Although the sequence $A^{n}:=P^{n}AP^{n}$ defined in Theorem \[thm\_short\] is positive and converges strongly to $A$, in general, it is not increasing in the Löwner order, so that Vigier’s Theorem does not apply, possibly suggesting why we only obtain convergence in the weak operator topology. With stronger assumptions on the operator $A$ and a well chosen selection of an ordered orthonormal basis of $H_{2}$, we conjecture that convergence in a stronger topology may be available. In particular, as a corollary to our main result, when $A$ is trace class, we establish in Corollary \[cor\_short\] that $$\mathcal{H}_{1}(A^{n}) \rightarrow \mathcal{H}_{1}(A) \, \, \text{in trace norm}\, .$$

For any $m \in H$, we let $m=(m_{1},m_{2})$ denote its decomposition in $H=H_{1}\oplus H_{2}$. Moreover, for any projection $Q:H\rightarrow H$ with $R(Q)=H_{2}$ we let $\hat{Q}: H_{1} \rightarrow H_{2}$ denote the unique operator such that $$Q= \begin{pmatrix}
  0& 0\\
\hat{Q}& 1
 \end{pmatrix}\, ,$$ and denote by $\hat{Q}^{*}:H_{2} \rightarrow H_{1}$ the adjoint of $\hat{Q}$ defined by the relation $\langle \hat{Q}^{*}h_{2}, h_{1}  \rangle_{H_{1}} =\langle h_{2}, \hat{Q}h_{1}  \rangle_{H_{2}}$ for all $h_{1} \in H_{1}, h_{2}\in H_{2}$.

The following theorem constitutes an expansion of our main result, Theorem \[thm\_main0\], to include natural approximations for the conditional covariance operator and the conditional expectation operator.

\[thm\_main\] Consider a Gaussian measure $\mu$ on an orthogonal direct sum $H=H_{1}\oplus H_{2}$ of separable Hilbert spaces with mean $m$ and covariance operator $C$. Then for all $t \in H_{2}$, the conditional measure $\mu_{t}$ is a Gaussian measure with covariance operator $\mathcal{H}_{1}(C).$

If the covariance operator $C$ is compatible with $H_{2}$, then for any oblique projection $Q$ in $\mathcal{P}(C,H_{2})\neq \emptyset$, the mean $m_{t}$ of the conditional measure $\mu_{t}$ is $$m_{t}= \begin{pmatrix}
  m_{1}+\hat{Q}^{*}(t-m_{2}) \\
t
 \end{pmatrix}\, .$$ In the general case, for any ordered orthonormal basis for $H_{2}$, let $H^{n}_{2}$ denote the span of the first $n$ basis elements, let $P^{n}:=P_{H_{1}}+P_{H^{n}_{2}}$ denote the orthogonal projection onto $H_{1}\oplus H^{n}_{2}$, and define the approximate $C^{n}:=P^{n}CP^{n}$. Then $C^{n}$ is compatible with $H_{2}$ for all $n$, and for any sequence $Q_{n} \in \mathcal{P}(C^{n},H_{2}) \neq \emptyset$ of oblique projections, we have $$m_{t}=  \begin{pmatrix}
  m_{1}+ \lim_{n\rightarrow\infty}{\hat{Q}_{n}^{*}(t-P_{H^{n}_{2}}m_{2})} \\
t
 \end{pmatrix}$$ for $\mu$-almost every $t$. If the sequence $Q_{n}$ eventually becomes the special element $Q_{n}=Q_{C^{n},H_{2}}$ defined near , then we have $$m_{t}=  \begin{pmatrix}
  m_{1}+ \lim_{n\rightarrow\infty}{\hat{Q}_{n}^{*}(t-m_{2})} \\
t
 \end{pmatrix}$$ for $\mu$-almost every $t$.

As a corollary to Theorem \[thm\_main\], we obtain a strengthening of the assertion of Theorem \[thm\_short\] when the operator $A$ is trace class.

\[cor\_short\] Consider the situation of Theorem \[thm\_short\] with $A$ trace class. Then $$\mathcal{H}_{1}(A^{n}) \rightarrow \mathcal{H}_{1}(A)\,\, \text{in trace norm}\, .$$

Proofs
======

First proof of Theorem \[thm\_main0\] {#sec_proof1}
-------------------------------------

Consider the Lebesgue-Bochner space $L^{2}(H,\mu,\mathcal{B}(H))$ space of (equivalence classes) of $H$-valued Borel measurable functions on $H$ whose squared norm $${\Vert f \Vert}^{2}_{L^{2}(H,\mu,\mathcal{B}(H))}:=\int_{H'}{{\Vert f(x) \Vert}^{2}_{H}d\mu(x)}$$ is integrable. For any square Bochner integrable function $f \in L^{2}(H,\mu,\mathcal{B}(H))$ and any $h \in H$, we have that $\langle f,h \rangle $ is square integrable, that is $\langle f,h \rangle \in L^{2}({\mathbb{R}},\mu,\mathcal{B}(H))$. Moreover, it is easy to see, see e.g.  [@AliprantisBorder:2006 Lem. 11.45], that if $f$ is Bochner integrable, then for all $h \in H$, we have $\langle f,h \rangle $ is Bochner integrable and $\int{\langle f,h \rangle d\mu}=\langle \int{fd\mu},h \rangle$.

Now consider the orthogonal decomposition $H=H_{1}\oplus H_{2}$ and the Borel ${\sigma}$-algebra $\mathcal{B}(H_{2})$. Let us denote the shorthand notation $$\mathcal{B}:=\mathcal{B}(H), \quad
\mathcal{B}_{2}:= \{ (H_{1},T):  T \in \mathcal{B}(H_{2})\}\, .$$ The definition of conditional expectation in Lebesgue-Bochner space, that is that ${\mathbb{E}}[f|\mathcal{B}_{2}]$ is the unique $\mu$-almost everywhere $\mathcal{B}_{2}$-measurable function such that $$\int_{B}{{\mathbb{E}}[f|\mathcal{B}_{2}]d\mu}=\int_{B}{fd\mu},\quad  B \in \mathcal{B}_{2}$$ combined with Hille’s theorem [@DiestelUhl Thm. II.6], that for each $h \in H$ we have $$\langle h,  \int_{B}{{\mathbb{E}}[f|\mathcal{B}_{2}]d\mu}\rangle =
\int_{B}{\langle h,{\mathbb{E}}[f|\mathcal{B}_{2}]\rangle d\mu}, \quad B \in \mathcal{B}_{2}$$ and $$\langle h, \int_{B}{f d\mu}\rangle = \int_{B}{\langle h, f \rangle d\mu}, \quad  B \in \mathcal{B}_{2}\, ,$$ implies that $${\mathbb{E}}[\langle h,f\rangle |\mathcal{B}_{2}]=  \langle h, {\mathbb{E}}[f|\mathcal{B}_{2}]\rangle,   \quad h \in H$$ thus implying the following commutative diagram for all $h \in H$: $$\label{fig0}
\begin{xy}
 (0,0)*+{L^{2}({\mathbb{R}},\mu,\mathcal{B})}="a";
(45,0)*+{L^{2}({\mathbb{R}},\mu,\mathcal{B}_{2})}="c";
 (0,30)*+{ L^{2}(H,\mu,\mathcal{B})}="a1";
(45,30)*+{ L^{2}(H,\mu,\mathcal{B}_{2})}="c1";
{\ar^{{\mathbb{E}}[\, \cdot\, |\mathcal{B}_{2}]} "a1";"c1"};
{\ar^{{\mathbb{E}}[\, \cdot\, |\mathcal{B}_{2}]} "a";"c"};
{\ar_{\langle h} "a1";"a"};
{\ar_{\langle h} "c1";"c"};
\end{xy}$$

When $\mu$ is a Gaussian measure, the theory of Gaussian Hilbert spaces, see e.g. Jansen [@janson1997gaussian], provides a stronger characterization of conditional expectation of the canonical random variable $X(h):=h, h \in H$ when conditioning on a subspace and captures the full linear nature of Gaussian conditioning. Let us assume henceforth that $\mu$ is a centered Gaussian measure. Then Fernique’s Theorem [@fernique1975regularite], see [@da2014stochastic Thm. 2.6], implies that the random variable $X$ is square Bochner integrable. For any element $h \in H$, let us denote the corresponding function $\xi_{h}:H \rightarrow {\mathbb{R}}$ defined by $\xi_{h}(h')=\langle h,h'\rangle, h' \in H$. Then the the discussion above shows that for any $h \in H$, that the real-valued random variable $\xi_{h} $ is square integrable, that is $\xi_{h} \in  L^{2}({\mathbb{R}},\mu,\mathcal{B})$, for all $h \in H$. Let $$\xi:H  \rightarrow  L^{2}({\mathbb{R}},\mu,\mathcal{B})$$ denote the resulting linear mapping defined by $$h \mapsto \xi_{h} \in  L^{2}({\mathbb{R}},\mu,\mathcal{B}), h \in H\, .$$ It is straightforward to show that $\xi$ is injective if and only if the covariance operator $C$ of the Gaussian measure $\mu$ is injective. By the definition of a centered Gaussian vector $X$, it follows that the law $(\xi_{h})_{*}\mu$ in ${\mathbb{R}}$ is a univariate centered Gaussian measure, that is $\xi_{h}$ is a centered Gaussian real-valued random variable. Consequently, let us consider the closed linear subspace $$H^{\mu}:={\mkern 1.5mu\overline{\mkern-1.5mu\xi(H)\mkern-1.5mu}\mkern 1.5mu} \subset  L^{2}({\mathbb{R}},\mu,\mathcal{B})$$ generated by the elements $\xi_{h}\in  L^{2}({\mathbb{R}},\mu,\mathcal{B}), h \in H$. By Jansen [@janson1997gaussian Thm. I.1.3], this closure $ H_{\mu} \subset  L^{2}({\mathbb{R}},\mu,\mathcal{B})$ also consists of centered Gaussian random variables, and since it is a closed subspace of a Hilbert space, it is a Hilbert space and therefore a Gaussian Hilbert space as defined in Jansen [@janson1997gaussian]. Moreover, by Jansen [@janson1997gaussian Thm. 8.15], $H_{\mu}$ is a feature space for the Cameron-Martin reproducing kernel Hilbert space with feature map $\xi:H \rightarrow H^{\mu}$ and reproducing kernel the covariance operator. For a closed Hilbert subspace, $H_{2}\subset H$, we can consider the closed linear subspace $$H_{2}^{\mu}:={\mkern 1.5mu\overline{\mkern-1.5mu\xi(H_{2})\mkern-1.5mu}\mkern 1.5mu} \subset  L^{2}({\mathbb{R}},\mu,\mathcal{B}_{2})$$ generated by the elements $\xi_{h_{2}},h_{2} \in H_{2}$ in the same way. $ H_{2}^{\mu}$ is also a Gaussian Hilbert space and we have the natural subspace identification $ H_{2}^{\mu} \subset H^{\mu} $. Since separable Hilbert spaces are Polish, and an orthonormal basis is a separating set, it follows, see e.g. Vakhania, Tarieladze and Chobanyan [@vakhania1987probability Thm. I.1.2], that for an orthonormal basis $e_{i},i \in I$ of a separable Hilbert space, that the ${\sigma}$-algebra generated by the corresponding real-valued functions ${\sigma}(\{\xi_{e_{i}},i\in I\})$ is the Borel ${\sigma}$-algebra of the Hilbert space. Consequently, we obtain from Janson [@janson1997gaussian Thm. 9.1] that for any $ h \in H$, that $$\begin{aligned}
{\mathbb{E}}[\xi_{h}|\mathcal{B}_{2}]&=&
{\mathbb{E}}[\xi_{h}|{\sigma}(\cup{\xi_{h_{2}}},h_{2} \in H_{2})]\\
&=& P_{H_{2}^{\mu}}\xi_{h}\end{aligned}$$ where $P_{H_{2}^{\mu}}:H^{\mu}\rightarrow H_{2}^{\mu}$ is orthogonal projection. That is, if we let ${\mathbb{E}}[\, \cdot|\mathcal{B}_{2}]: L^{2}({\mathbb{R}},\mu,\mathcal{B})  \rightarrow
 L^{2}({\mathbb{R}},\mu,\mathcal{B}_{2})
$ be the conditional expectation represented as orthogonal projection and ${\mathbb{E}}[\, \cdot|\mathcal{B}_{2}]:  H^{\mu} \rightarrow H_{2}^{\mu}$ be the conditional expectation represented as orthogonal projection from the linear subspace $H^{\mu} \subset L^{2}({\mathbb{R}},\mu,\mathcal{B})$ onto the closed subspace $ H_{2}^{\mu} \subset H^{\mu} $, we have the following commutative diagram, where $i_{H^{\mu}}:H^{\mu} \rightarrow
L^{2}({\mathbb{R}},\mu,\mathcal{B})$ and $i_{H_{2}^{\mu}}:H_{2}^{\mu} \rightarrow
L^{2}({\mathbb{R}},\mu,\mathcal{B}_{2})$ denote the closed subspace injections. $$\label{fig1}
\begin{xy}
 (0,0)*+{H^{\mu}}="a";
(45,0)*+{H_{2}^{\mu}}="c";
 (0,30)*+{ L^{2}({\mathbb{R}},\mu,\mathcal{B})}="a1";
(45,30)*+{ L^{2}({\mathbb{R}},\mu,\mathcal{B}_{2})}="c1";
{\ar^{{\mathbb{E}}[\, \cdot\, |\mathcal{B}_{2}]} "a1";"c1"};
{\ar^{{\mathbb{E}}[\, \cdot\, |\mathcal{B}_{2}]} "a";"c"};
{\ar^{i_{H^{\mu}}} "a";"a1"};
{\ar^{i_{H_{2}^{\mu}}} "c";"c1"};
\end{xy}$$ which when combined with Figure \[fig0\], representing the commutativity of vector projection and conditional expectation, produce the following commutative diagram for all $h \in H$: $$\label{fig2}
\begin{xy}
(0,-30)*+{H}="a0";
(45,-30)*+{H_{2}}="c0";
(0,0)*+{H^{\mu}}="a";
(45,0)*+{H_{2}^{\mu}}="c";
 (0,30)*+{ L^{2}({\mathbb{R}},\mu,\mathcal{B})}="a1";
(45,30)*+{ L^{2}({\mathbb{R}},\mu,\mathcal{B}_{2})}="c1";
{\ar^{{\mathbb{E}}[\, \cdot\, |\mathcal{B}_{2}]} "a1";"c1"};
{\ar^{{\mathbb{E}}[\, \cdot\, |\mathcal{B}_{2}]} "a";"c"};
{\ar^{i_{H^{\mu}}} "a";"a1"};
{\ar^{i_{H_{2}^{\mu}}} "c";"c1"};
 (0,60)*+{ L^{2}(H,\mu,\mathcal{B})}="a2";
(45,60)*+{ L^{2}(H,\mu,\mathcal{B}_{2})}="c2";
{\ar^{{\mathbb{E}}[\, \cdot\, |\mathcal{B}_{2}]} "a2";"c2"};
{\ar_{\langle h} "a2";"a1"};
{\ar_{\langle h} "c2";"c1"};
{\ar^{\xi} "a0";"a"};
{\ar^{\xi} "c0";"c"};
\end{xy}$$ Although there is a natural projection map $P_{H_{2}}:H \rightarrow  H_{2}$ for the bottom of this diagram, in general it cannot be inserted here and maintain the commutativity of the diagram. This comes from the fact that there may exist an $h \in H$ such that $\xi_{h}=0$. However, this does not imply that $\xi_{P_{H_{2}}h}=0$.

We are now prepared to obtain the main assertion. The covariance operator of the random variable $X$ is defined by $$\begin{aligned}
 \bigl\langle C h, h' \bigr\rangle&=&{\mathbb{E}}_{\mu}\bigl[\langle X,h\rangle \langle X,h'\rangle\bigr]\\
&=&{\mathbb{E}}_{\mu}\bigl[\xi_{h}\xi_{h'}\bigr], \quad  h,h'
\in H\, .
 \end{aligned}$$ Moreover, by the theorem of normal correlation and the commutativity of the diagram , the conditional covariance operator is defined by $$\begin{aligned}
\bigl\langle C(X|X_{2}) h, h' \bigr\rangle&=&{\mathbb{E}}_{\mu}\Bigl[\bigl\langle X- {\mathbb{E}}[X|\mathcal{B}_{2}],h
\bigr\rangle
\bigl\langle  X-{\mathbb{E}}[X|\mathcal{B}_{2}] ,h'\bigr\rangle|\mathcal{B}_{2}\Bigr]
\\
&=&{\mathbb{E}}_{\mu}\Bigl[\bigl\langle X- {\mathbb{E}}[X|\mathcal{B}_{2}],h
\bigr\rangle
\bigl\langle  X-{\mathbb{E}}[X|\mathcal{B}_{2}] ,h'\bigr\rangle \Bigr]\\
&=&{\mathbb{E}}_{\mu}\Bigl[\bigl(\xi_{h}-{\mathbb{E}}[\xi_{h}|\mathcal{B}_{2}]\bigr)
\bigl(\xi_{h'}-{\mathbb{E}}[\xi_{h'}|\mathcal{B}_{2}]\bigr)\Bigr],\quad
h,h'
\in H\, .\end{aligned}$$ In terms of the Gaussian Hilbert spaces $H^{\mu}_{2}\subset H^{\mu}$, using the commutativity of the diagram and the identification of the conditional expectation with orthogonal projection, we conclude that $$\label{id_cov}
\bigl\langle C h, h' \bigr\rangle=\langle \xi_{h},\xi_{h'}\rangle_{H^{\mu}}, \quad h, h'
\in H\,$$ and $$\label{id_condcov}
\bigl\langle C(X|X_{2}) h, h' \bigr\rangle=
\langle (I-P_{H_{2}^{\mu}}) \xi_{h},(I-P_{H_{2}^{\mu}})\xi_{h'}\rangle_{H^{\mu}}, \quad h, h'
\in H \, .$$ Since the orthogonal projection $P_{H_{2}^{\mu}}$ is a metric projection of $H^{\mu}$ onto $H_{2}^{\mu}$, we can express the dual optimization problem to the metric projection as follows: for any $h \in H$, using the decomposition $h=h_{1}+h_{2}$ with $h_{1}\in H_{1}, h_{2} \in H_{2}$, we decompose $\xi_{h}=\xi_{h_{1}+h_{2}}=\xi_{h_{1}}+\xi_{h_{2}}$. Then, noting that $(I-P_{H_{2}^{\mu}})\xi_{h_{2}}=0$, we obtain $$\begin{aligned}
 {\Vert \xi_{h} \Vert}^{2}_{H^{\mu}}&=& {\Vert \xi_{h_{1}}+\xi_{h_{2}} \Vert}^{2}_{H^{\mu}}\\
&=& {\Vert (I-P_{H_{2}^{\mu}})(\xi_{h_{1}}+\xi_{h_{2}}) \Vert}^{2}_{H^{\mu}}+ {\Vert P_{H_{2}^{\mu}}(\xi_{h_{1}}+\xi_{h_{2}}) \Vert}^{2}_{H^{\mu}}\\
&=& {\Vert (I-P_{H_{2}^{\mu}})\xi_{h_{1}} \Vert}^{2}_{H^{\mu}}+ {\Vert P_{H_{2}^{\mu}}\xi_{h_{1}}+\xi_{h_{2}} \Vert}^{2}_{H^{\mu}}\, .\end{aligned}$$ Since in the second term on the right-hand side $P_{H_{2}^{\mu}}\xi_{h_{1}} \in H_{2}^{\nu}$, there is a sequence $ h_{2}^{n}, n=1,...$ such that the corresponding sequence $\xi_{h_{2}^{n}}$ converges to $-P_{H_{2}^{\mu}}\xi_{h_{1}}$ in $ L^{2}({\mathbb{R}},\mu,\mathcal{B})$ and therefore $H^{\mu}$, we conclude that $${\Vert (I-P_{H_{2}^{\mu}})\xi_{h_{1}} \Vert}^{2}_{H^{\mu}}=\inf_{h_{2}\in H_{2}}{{\Vert \xi_{h_{1}}+\xi_{h_{2}} \Vert}^{2}_{H^{\mu}}}\,  \, .$$ From the identifications and , we conclude that $$\bigl\langle C(X|X_{2}) h_{1}, h_{1} \bigr\rangle =\inf_{h_{2}\in H_{2}}{\bigl\langle C(X)(h_{1}+h_{2}),
h_{1}+ h_{2} \bigr\rangle},\, .$$ Therefore, Anderson and Trapp [@anderson1975shorted Thm. 6] implies the assertion $$C(X|X_{2})=\mathcal{H}_{1}\bigl(C\bigr)\, .$$ The assertion in the non-centered case follows by simple translation.

Proof of Theorem \[thm\_short\]
-------------------------------

Since $P_{H_{2}}A^{n}P_{H_{2}} =P_{H_{2}^{n}}A^{n}P_{H_{2}^{n}}$, the range of $P_{H_{2}}A^{n}P_{H_{2}}$ is finite dimensional, and therefore closed, so that it follows from Corach, Maestripieri and Stojanoff [@corach2006projections Lem. 3.8] that $A^{n}$ is compatible with $H_{2}$ for all $n$.

Now we utilize the approximation results of Butler and Morley [@butler1988note] for the shorted operator. By [@butler1988note Lem. 1], for $c \in H$ and for fixed $n$, it follows that there exists a sequence $y_{m}^{n}\in H_{2}, m=1,\ldots$ and a real number $M$ such that $$\begin{array}{cccc}
A^{n}_{11}c+A^{n}_{12}y^{n}_{m} & \rightarrow &\mathcal{H}_{1}(A^{n})c\, , &  \qquad   m \rightarrow \infty\\
A^{n}_{21}c+A^{n}_{22}y^{n}_{m} & \rightarrow &  0\, ,  &   \qquad  m \rightarrow \infty\\
\langle A^{n}_{22}y^{n}_{m}, y^{n}_{m} \rangle & \leq &  M\, ,  & \qquad  \forall m \, .
\end{array}$$ Since $A^{n}_{11}=A_{11}$, $A^{n}_{12}= A_{12}P_{H_{2}^{n}}$, $A^{n}_{21}= P_{H_{2}^{n}}A_{21}$, and $A^{n}_{22}= P_{H_{2}^{n}}A_{22}P_{H_{2}^{n}}$ this can be written as $$\begin{array}{cccc}
A_{11}c+A_{12}P_{H_{2}^{n}}y^{n}_{m} & \rightarrow & \mathcal{H}_{1}(A^{n})c \, , & \qquad   m \rightarrow \infty\\
P_{H_{2}^{n}}A_{21}c+P_{H_{2}^{n}}A_{22}P_{H_{2}^{n}}y^{n}_{m} & \rightarrow &  0\, , &   \qquad  m \rightarrow \infty\\
\langle A_{22}P_{H_{2}^{n}}y^{n}_{m}, P_{H_{2}^{n}}y^{n}_{m} \rangle & \leq &  M\, , &\qquad  \forall m\, .
\end{array}$$ Since these equations only depend on $P_{H_{2}^{n}}y^{n}_{m}$ we can further assume that $P_{(H_{2}^{n})^{\perp}}y^{n}_{m}=0, \, m =1,\ldots,$ where $P_{(H_{2}^{n})^{\perp}}$ is the orthogonal projection onto $(H_{2}^{n})^{\perp} \subset H_{2}$. That is, we can assume that $P_{H_{2}^{n}}y^{n}_{m}=y^{n}_{m}, m=1,\ldots$ and therefore $$\label{eq_zzzz}
\begin{array}{cccc}
A_{11}c+A_{12}y^{n}_{m} & \rightarrow & \mathcal{H}_{1}(A^{n})c \, , & \qquad   m \rightarrow \infty\\
P_{H_{2}^{n}}A_{21}c+P_{H_{2}^{n}}A_{22}y^{n}_{m} & \rightarrow &  0\, , &   \qquad  m \rightarrow \infty\\
\langle A_{22}y^{n}_{m}, y^{n}_{m} \rangle & \leq &  M\, , &\qquad  \forall m\, .
\end{array}$$ It follows from $\mathcal{H}_{1}(A^{n}) \leq A^{n}$ that ${\Vert \sqrt{\mathcal{H}_{1}(A^{n})} \Vert}\leq {\Vert \sqrt{A^{n}} \Vert}$ for the unique square root, guaranteed to exist by Riesz and Sz.-Nagy [@riesz1955functional Sec. 104]. Consequently, Conway [@conway1990course Prop. II.2.7] implies that ${\Vert \mathcal{H}_{1}(A^{n}) \Vert}\leq {\Vert A^{n} \Vert}$ for all $n$ and since ${\Vert A^{n} \Vert}\leq {\Vert A \Vert}$ for all $n$ it follows that ${\Vert \mathcal{H}_{1}(A^{n}) \Vert}\leq {\Vert A \Vert}$ for all $n$. Consequently, the sequence $\mathcal{H}_{1}(A^{n})c$ is bounded. Therefore there exists a weakly convergent subsequence. Let $n'$ denote the index of any weakly convergent subsequence, so that $$\label{eq_bbbb}
\mathcal{H}_{1}(A^{n'})c \xrightarrow{\omega} d',\quad n' \rightarrow \infty \,$$ for some $d'$ depending on the subsequence. Now the strong convergence of the lefthand side to the righthand side in is maintained for the subsequence $n'$ and, since for the subsequence the first term on the righthand side converges weakly to $d'$, it follows that we can define a monotonically increasing function $m(n')$ and use it to define a new sequence $\hat{y}^{n'}:=y^{n'}_{m(n')}$ such that $$\label{eq_zzzz2}
\begin{array}{cccc}
A_{11}c+A_{12}\hat{y}^{n'} & \xrightarrow{\omega} & d'\, , & \qquad   n' \rightarrow \infty\\
P_{H_{2}^{n'}}A_{21}c+P_{H_{2}^{n'}}A_{22}\hat{y}^{n'}  & \rightarrow &  0\, , &   \qquad  n' \rightarrow \infty\\
\langle A_{22}\hat{y}^{n'} , \hat{y}^{n'}  \rangle & \leq &  M\, , &\qquad  \forall n'\, .
\end{array}$$ Since $P_{H_{2}^{n}}$ is strongly convergent to $P_{H_{2}}$ it follows that $P_{H_{2}^{n'}}$ is strongly convergent to $P_{H_{2}}$, so that $P_{H_{2}^{n'}}A_{21}c$ converges to $A_{21}c$ and $P_{H_{2}^{n'}}A_{22}\hat{y}^{n'}$ converges to $-A_{21}c$. Moreover, by Reid’s inequality [@reid1951symmetrizable Cor. 2] we have $$\label{ineq_reid}
  {\Vert A_{22}\hat{y}^{n'} \Vert}^{2}_{H_{2}}\leq {\Vert A_{22} \Vert}\langle A_{22}\hat{y}^{n'}, \hat{y}^{n'}\rangle
\leq {\Vert A_{22} \Vert} M\, ,$$ for all $n'$, so that the sequence $A_{22}\hat{y}^{n'}$ is bounded. Since weak convergence of a bounded sequence on a separable Hilbert space is equivalent to the convergence with respect to each element of any orthonormal basis, it follows that $A_{22}\hat{y}^{n'}$ is weakly convergent to $-A_{21}c$. From , we obtain $$\label{eq_abc}
\begin{array}{cccc}
A_{12}\hat{y}^{n'} & \xrightarrow{\omega} & d'-A_{11}c\, , & \qquad   n' \rightarrow \infty\\
A_{22}\hat{y}^{n'}& \xrightarrow{\omega} & -A_{21}c \, , &  \qquad  n' \rightarrow \infty\, .
\end{array}$$

From Kakutani’s [@kakutani1938weak] generalization of the Banach-Saks Theorem it follows that we can select a subsequence $\grave{n}$ of $n'$ such that the Cesaro means of $A_{22}\hat{y}^{\grave{n}}$ and $A_{12}\hat{y}^{\grave{n}}  $ converge strongly in . That is, if we consider the Cesaro means $$z^{\grave{n}}=\frac{1}{\grave{n}}\sum_{i=1}^{\grave{n}}{\hat{y}^{\grave{n}}}$$ we have $$\begin{array}{cccc}
A_{12}z^{n'} & \rightarrow & d'-A_{11}c\, , & \qquad   n' \rightarrow \infty\\
A_{22}z^{n'}& \rightarrow & -A_{21}c \, , &  \qquad  n' \rightarrow \infty \, .
\end{array}$$ Since $A_{22}\geq 0$ it follows that the function $y \mapsto \langle A_{22}y,y\rangle$ is convex, so that $\langle A_{22} z^{\grave{n}}, z^{\grave{n}} \rangle \leq M $ for all $\grave{n}$, so that $$\label{eq_zzzz3}
\begin{array}{cccc}
A_{11}c+A_{12}z^{\grave{n}} & \rightarrow & d'\, , & \qquad   \grave{n} \rightarrow \infty\\
A_{21}c+A_{22}z^{\grave{n}}& \rightarrow & 0 \, , &  \qquad  \grave{n} \rightarrow \infty\\
\langle A_{22}z^{\grave{n}}, z^{\grave{n}} \rangle & \leq & M\, , & \qquad  \forall \grave{n}\, .
\end{array}$$ It therefore follows from the from the main result of Butler and Morley [@butler1988note Thm. 1] that $$d'  =\mathcal{H}_{1}(A)c\, .$$ Consequently, by , we obtain that $$\label{eq_bbbb2}
\mathcal{H}_{1}(A^{n'})c \xrightarrow{\omega} \mathcal{H}_{1}(A)c,\quad n' \rightarrow \infty \, .$$ Since this limit is independent of the chosen weakly converging subsequence, it follows, see e.g. Zeidler [@zeidler1989nonlinear Prop. 10.13], that the full sequence weakly converges to the same limit, that is we have $$\label{eq_bbbb3}
\mathcal{H}_{1}(A^{n})c \xrightarrow{\omega} \mathcal{H}_{1}(A)c,\quad n \rightarrow \infty \, ,$$ and since $c$ was arbitrary we conclude that $$\mathcal{H}_{1}(A)= \omega\mbox{-}\\lim_{n\rightarrow\infty}{ \mathcal{H}_{1}(A^{n})}\, .$$

Proof of Theorem \[thm\_main\]
------------------------------

Let us first establish the assertion when $C$ is compatible with $H_{2}$. Consider the operator $\hat{C}:H \rightarrow H$ defined by $$\hat{C}:=\mathcal{H}_{1}(C) +P_{2} CP_{2}\, .$$ Since $C$ is compatible with $H_{2}$, there exists an oblique projection $Q \in \mathcal{P}(C,H_{2})$, and Corach, Maestripieri and Stojanoff [@corach2001schur Prop. 4.2] asserts that for $E:=1-Q$, we have $$\label{e1}
\mathcal{H}_{1}(C)=CE=E^{*}CE\, .$$ Since $Q^{*}C=CQ$ it follows that $E^{*}C=CE$, and since $Q$ is a projection, it follows that $QE=EQ=0$ and that $E$ is a projection. Moreover, since $R(Q)=H_{2}$ it follows that $ker(E)=H_{2}$, so that we obtain $P_{2}Q=Q$ and $EP_{1}=E$ and therefore $Q^{*}P_{2}=Q^{*}$ and $P_{1}E^{*}=E^{*}$. Consequently, we obtain $$\begin{aligned}
 (P_{1}+Q)^{*}\hat{C}(P_{1}+Q)&=& (P_{1}+Q)^{*}(E^{*}CE+P_{2}CP_{2})(P_{1}+Q)\\
&=& (P_{1}+Q)^{*}(E^{*}CE+P_{2}CQ)\\
&=&E^{*}CE+ Q^{*}CQ \\
&=& CE+CQ \\
&=& C\, ,\end{aligned}$$ that is, $$\label{id_covariance}
C=(P_{1}+Q)^{*}\hat{C}(P_{1}+Q)\, .$$

Since $Q$ is a projection onto $H_{2}$, it follows that $P_{1}+Q$ is lower triangular in its partitioned representation and therefore the fundamental pivot produces an explicit, and most importantly continuous, inverse. Indeed, if we use the partition representation $$Q= \begin{pmatrix}
  0& 0\\
\hat{Q}& 1
 \end{pmatrix}\, ,$$ we see that $$(P_{1}+Q)= \begin{pmatrix}
  1 & 0  \\
   \hat{Q} & 1
 \end{pmatrix}$$ from which we conclude that $$(P_{1}+Q)^{-1}= \begin{pmatrix}
  1 &  0  \\
   -\hat{Q} & 1
 \end{pmatrix}$$ Without partitioning, using $P_{1}Q=0$ and $QP_{2}=P_{2}$, we obtain $$\begin{aligned}
(2-P_{1}-Q)(P_{1}+Q) &=&2P_{1}+2Q -(P_{1}^{2}+P_{1}Q+QP_{1}+Q^{2})\\
&=&2P_{1} +2Q- P_{1} -P_{1}Q-QP_{1}-Q\\
&=&P_{1}+Q-QP_{1}\\
&=&P_{1}+QP_{2}\\
&=&P_{1} +P_{2}\\
&=& 1\end{aligned}$$ and so confirm that $$\label{eq_inv}
(P_{1}+Q)^{-1}=2-P_{1}-Q\, .$$

Following the proof of Hairer, Stuart, Voss, and Wiber [@hairer2005analysis Lem. 4.3], let $\mathcal{N}(m,C)$ denote the Gaussian measure with mean $m$ and covariance operator $C$ and consider the transformation $$(P_{1}+Q)^{-*}:H \rightarrow H\, ,$$ where we use the notation $A^{-*}$ for $(A^{-1})^{*}=(A^{*})^{-1}$. From we obtain $$\label{covariance2}
(P_{1}+Q)^{-*}C(P_{1}+Q)^{-1}=\hat{C}$$ so that the transformation law for Gaussian measures, see Maniglia and Rhandi [@maniglia2004gaussian Ch. 1, Lem. 1.2.7], implies that $$\bigl((P_{1}+Q)^{-*}\bigr)_{*}\mathcal{N}(m,C)=\mathcal{N}\bigl((P_{1}+Q)^{-*}m,\hat{C}\bigr)\, .$$

Since $$(P_{1}+Q)^{-1}= \begin{pmatrix}
  1 &  0  \\
   -\hat{Q} & 1
 \end{pmatrix}$$ we obtain $$(P_{1}+Q)^{-*}
= \begin{pmatrix}
  1 &  -\hat{Q}^{*}  \\
   0 & 1
 \end{pmatrix}$$ and therefore $$(P_{1}+Q)^{-*}m=
\begin{pmatrix}
  m_{1}-\hat{Q}^{*}m_{2}  \\
  m_{2}
 \end{pmatrix}\, .$$ Since the partition representation of $\hat{C}$ is $$\hat{C}=  \begin{pmatrix}
(\mathcal{H}_{1}(C))_{11}  &  0  \\
  0 &  C_{22}
 \end{pmatrix}$$ the components of the corresponding Gaussian random variable are uncorrelated and therefore independent. That is, we have $$\mathcal{N}\bigl((P_{1}+Q)^{-*}m,\hat{C}\bigr)=
\mathcal{N}\bigl(m_{1}-\hat{Q}^{*}m_{2},(\mathcal{H}_{1}(C))_{11}\bigr)
\mathcal{N}\bigl(m_{2},C_{22}\bigr)
\, .$$ This independence facilitates the computation of the conditional measure as follows. Let $X=(X_{1},X_{2})$ denote the random variable associated with the Gaussian measure $\mathcal{N}(m,C)$ and consider the transformed random variable $Y=(P_{1}+Q)^{-*}X$ with the product law $\mathcal{N}\bigl(m_{1}-\hat{Q}^{*}m_{2},(\mathcal{H}_{1}(C))_{11}\bigr)\mathcal{N}\bigl(m_{2},C_{22}\bigr)$. Then, $$\begin{aligned}
Y_{1}&=& X_{1}-\hat{Q}^{*}X_{2}\\
Y_{2}&=&X_{2}\end{aligned}$$ can be used to compute the conditional expectation as $$\begin{aligned}
{\mathbb{E}}[X_{1}|X_{2}]&=&{\mathbb{E}}[X_{1}-\hat{Q}^{*}X_{2}|X_{2}]+{\mathbb{E}}[\hat{Q}^{*}X_{2}|X_{2}]\\
&=&{\mathbb{E}}[Y_{1}|Y_{2})]+{\mathbb{E}}[\hat{Q}^{*}X_{2}|X_{2}]\\
&=&{\mathbb{E}}[Y_{1}]+\hat{Q}^{*}X_{2}\, ,\end{aligned}$$ obtaining $${\mathbb{E}}[X_{1}|X_{2}]=
{\mathbb{E}}[Y_{1}]+\hat{Q}^{*}X_{2}\, ,$$ so that we conclude that $${\mathbb{E}}[X_{1}|X_{2}]=m_{1}+\hat{Q}^{*}(X_{2}-m_{2})\, .$$ A similar calculation obtains the covariance $$\label{id_cov}
C(X|X_{2})=\mathcal{H}_{1}(C),$$ thus establishing the assertion in the compatible case.

For the general case, we do not assume that $C$ is compatible with $H_{2}$. Consider an ordered orthonormal basis for $H_{2}$, let $H^{n}_{2}$ denote the span of the first $n$ basis elements, let $P^{n}:=P_{H_{1}}+P_{H^{n}_{2}}$ denote the orthogonal projection onto $H_{1}\oplus H^{n}_{2}$, and consider the sequence of Gaussian measures $\mu_{n}:=P^{n}_{*}\mu$ with the mean $P^{n}m$ and covariance operators $$C^{n}:=P^{n}CP^{n},\quad n=1,\ldots$$ As asserted in Theorem \[thm\_short\], $C^{n}$ is compatible with $H_{2}$ for all $n$, and the sequence $\mathcal{H}_{1}(C^{n})$ converges weakly to $\mathcal{H}_{1}(C)$. Let $C(X_{1}|X_{2}^{n})$ and $C(X_{1}|X_{2})$ denote the conditional covariance operators associated with the measure $\mu$. Then we will show that $C(X_{1}|X_{2}^{n})=\mathcal{H}_{1}(C^{n})$, so that the assertion regarding the conditional covariance operators is established if we demonstrate that the sequence of conditional covariance operators $C(X_{1}|X_{2}^{n})$ converges weakly to $C(X_{1}|X_{2})$.

To both ends, consider the Lebesgue-Bochner space $L^{2}(H,\mu,\mathcal{B})$ space of (equivalence classes) of $H$-valued Borel measurable functions on $H$ whose squared norm $${\Vert f \Vert}^{2}_{L^{2}(H,\mu,\mathcal{B})}:=\int_{H}{{\Vert f(x) \Vert}^{2}_{H}d\mu(x)}$$ is integrable. Since Fernique’s Theorem [@fernique1975regularite], see [@da2014stochastic Thm. 2.6], implies that the random variable $X$ is square Bochner integrable, it follows that the Gaussian random variables $P^{n}X$ are also square Bochner integrable with respect to $\mu$. Let us denote $\mathcal{B}_{2}:= \{ (H_{1},T):  T \in \mathcal{B}(H_{2})\}$ and $\mathcal{B}^{n}_{2}:=\{ (H_{1},T^{n},(H_{2}^{n})^{\perp}):  T^{n} \in \mathcal{B}(H^{n}_{2})\}$, and let $\mu_{n}:=P^{n}_{*}\mu$ denote the image under the projection. $\mu_{n}$ is a Gaussian measure on $H$ with mean $P^{n}m$ and covariance $C^{n}$.

Now consider a function $f:H \rightarrow H$ which is Bochner square integrable with respect to $\mu$ and satisfies $f\circ P^{n}=f$. Then, using the change of variables formula for Bochner integrals, see Bashirov [@bashirov2003partially Thm. 2.26], along with the fact that $(P^{n})^{-1}\mathcal{B}_{2}=\mathcal{B}^{n}_{2}$, and using the fact that for an arbitrary $\mathcal{B}^{n}_{2}$-measurable function $g$ we have $g=g\circ P^{n}$, it follows that for $ A \in \mathcal{B}_{2}$, we have $$\begin{aligned}
 \int_{A}{fd\mu_{n}}&=& \int_{(P^{n})^{-1}A}{f\circ P^{n}d\mu}\\
&=& \int_{(P^{n})^{-1}A}{fd\mu}\\
&=&\int_{(P^{n})^{-1}A}{{\mathbb{E}}_{\mu}\bigl[f|(P^{n})^{-1}\mathcal{B}_{2}\bigr]d\mu}\\
&=&\int_{(P^{n})^{-1}A}{{\mathbb{E}}_{\mu}\bigl[f|\mathcal{B}^{n}_{2}\bigr]d\mu}\\
&=&\int_{(P^{n})^{-1}A}{{\mathbb{E}}_{\mu}\bigl[f|\mathcal{B}^{n}_{2}\bigr]\circ P^{n}d\mu}\\
&=&\int_{A}{{\mathbb{E}}_{\mu}\bigl[f|\mathcal{B}^{n}_{2}\bigr]d\mu_{n}}\\\end{aligned}$$ we obtain $$\label{eq_martingale}
 {\mathbb{E}}_{\mu_{n}}\bigl[f|\mathcal{B}_{2}\bigr]=  {\mathbb{E}}_{\mu}\bigl[f|\mathcal{B}^{n}_{2}\bigr]\, ,$$ and conclude that the sequence $ {\mathbb{E}}_{\mu_{n}}\bigl[f|\mathcal{B}_{2}\bigr], n=1 \ldots $ is a martingale corresponding to the increasing family of ${\sigma}$-algebras $\mathcal{B}^{n}_{2}$. Moreover, it is easy to see that holds for real valued functions $f:H \rightarrow R$ which are square integrable with respect to $\mu$ and satisfy $f\circ P^{n}=f$. With the choice $f:=X_{1}$, we clearly have $X_{1}\circ P^{n}=X_{1}$, so that if we denote $X^{n}_{2}:=P^{n}X_{2}$, we conclude that the sequence $$\label{id_martingale}
{\mathbb{E}}_{\mu_{n}}[X_{1}|X_{2}]={\mathbb{E}}_{\mu}[X_{1}|X^{n}_{2}]\, ,\quad  n=1,\ldots$$ is a martingale. Since conditional expectation is a contraction, it follows that the $L^{2}$ norm of all the conditional expectations are uniformly bounded by the $L^{2}$ norm of $X$. Then by the Martingale Convergence Theorem of Diestel and Uhl , ${\mathbb{E}}_{\mu_{n}}\bigl[X_{1}|X_{2}\bigr]$ converges to ${\mathbb{E}}_{\mu}\bigl[X_{1}|X_{2}\bigr]$ in $L^{2}(H,\mu,\mathcal{B})$.

For the conditional covariance operators, observe that implies that $$\label{id_martingale2}
X-{\mathbb{E}}_{\mu_{n}}[X|X_{2}]= X_{1}-{\mathbb{E}}_{\mu}[X_{1}|X^{n}_{2}]\,$$ for all $n$, so that for $h_{1}, h_{2} \in H$, we have $$\begin{aligned}
\langle C_{\mu_{n}}(X|X_{2})h_{1},h_{2}\rangle
&:=&{\mathbb{E}}_{\mu_{n}}\Bigl[\bigl\langle X-{\mathbb{E}}_{\mu_{n}}[X|X_{2}], h_{1}\bigr\rangle\bigl\langle X-{\mathbb{E}}_{\mu_{n}}[X|X_{2}] ,  h_{2}\bigr\rangle\big|X_{2}\Bigr]\\
&=&{\mathbb{E}}_{\mu_{n}}\Bigl[\bigl\langle X_{1}-{\mathbb{E}}_{\mu}[X_{1}|X^{n}_{2}], h_{1}\bigr\rangle\bigl\langle X_{1}
-{\mathbb{E}}_{\mu}[X_{1}|X^{n}_{2}] ,
h_{2}\bigr\rangle\big|X_{2}\Bigr]\end{aligned}$$ and since the integrand $ f:=\bigl\langle X_{1}-{\mathbb{E}}_{\mu}[X_{1}|X^{n}_{2}], h_{1}\bigr\rangle\bigl\langle X_{1}
-{\mathbb{E}}_{\mu}[X_{1}|X^{n}_{2}] ,
h_{2}\bigr\rangle$ satisfies $f\circ P^{n}=f$, it follows from that $$\begin{aligned}
&&{\mathbb{E}}_{\mu_{n}}\Bigl[\bigl\langle X_{1}-{\mathbb{E}}_{\mu}[X_{1}|X^{n}_{2}], h_{1}\bigr\rangle\bigl\langle X_{1}
-{\mathbb{E}}_{\mu}[X_{1}|X^{n}_{2}] ,
h_{2}\bigr\rangle\big|X_{2}\Bigr]\\
&=&{\mathbb{E}}_{\mu}\Bigl[\bigl\langle X_{1}-{\mathbb{E}}_{\mu}[X_{1}|X^{n}_{2}], h_{1}\bigr\rangle\bigl\langle X_{1}
-{\mathbb{E}}_{\mu}[X_{1}|X^{n}_{2}] ,
h_{2}\bigr\rangle\big|X^{n}_{2}\Bigr]\end{aligned}$$ so that using the theorem of normal correlation, we obtain $$\label{id_cov33}
\langle C_{\mu_{n}}(X|X_{2})h_{1},h_{2}\rangle={\mathbb{E}}_{\mu}\Bigl[\bigl\langle X_{1}-{\mathbb{E}}_{\mu}[X_{1}|X^{n}_{2}], h_{1}\bigr\rangle\bigl\langle X_{1}
-{\mathbb{E}}_{\mu}[X_{1}|X^{n}_{2}] ,
h_{2}\bigr\rangle\Bigr]\, .$$ Since the theorem of normal correlation also shows that $$\begin{aligned}
\langle C_{\mu}(X|X_{2})h_{1},h_{2}\rangle&:=&
{\mathbb{E}}_{\mu}\Bigl[\bigl\langle X-{\mathbb{E}}_{\mu}[X|X_{2}], h_{1}\bigr\rangle\bigl\langle X-{\mathbb{E}}_{\mu}[X|X_{2}] ,  h_{2}\bigr\rangle \big|X_{2}\Bigr]\\
&=&{\mathbb{E}}_{\mu}\Bigl[\bigl\langle X-{\mathbb{E}}_{\mu}[X|X_{2}], h_{1}\bigr\rangle\bigl\langle X-{\mathbb{E}}_{\mu}[X|X_{2}] ,  h_{2}\bigr\rangle \Bigr]\\
&=&
{\mathbb{E}}_{\mu}\Bigl[\bigl\langle X_{1}-{\mathbb{E}}_{\mu}[X_{1}|X_{2}], h_{1}\bigr\rangle\bigl\langle X_{1}-{\mathbb{E}}_{\mu}[X_{1}|X_{2}] ,  h_{2}\bigr\rangle \Bigr]\, ,\end{aligned}$$ the difference in the covariances can be decomposed as $$\begin{aligned}
&&\langle C_{\mu_{n}}(X_{1}|X_{2})h_{1},h_{2}\rangle-\langle C_{\mu}(X_{1}|X_{2})h_{1},h_{2}\rangle \\
&=& {\mathbb{E}}_{\mu}\Bigl[\bigl\langle X_{1}-{\mathbb{E}}_{\mu}[X_{1}|X^{n}_{2}], h_{1}\bigr\rangle\bigl\langle X_{1}-{\mathbb{E}}_{\mu}[X_{1}|X^{n}_{2}] ,  h_{2}\bigr\rangle \Bigr]\\
&&-{\mathbb{E}}_{\mu}\Bigl[\bigl\langle X_{1}-{\mathbb{E}}_{\mu}[X_{1}|X_{2}], h_{1}\bigr\rangle\bigl\langle X_{1}-{\mathbb{E}}_{\mu}[X_{1}|X_{2}] ,  h_{2}\bigr\rangle \Bigr]\\
&=& {\mathbb{E}}_{\mu}\Bigl[\bigl\langle {\mathbb{E}}_{\mu}[X_{1}|X_{2}]- {\mathbb{E}}_{\mu}[X_{1}|X^{n}_{2}], h_{1}\bigr\rangle\bigl\langle X_{1},  h_{2}\bigr\rangle \Bigr]+
{\mathbb{E}}_{\mu}\Bigl[\bigl\langle X_{1},h_{1}\bigr\rangle\bigl\langle {\mathbb{E}}_{\mu}[X_{1}|X_{2}]- {\mathbb{E}}_{\mu}[X_{1}|X^{n}_{2}] ,  h_{2}\bigr\rangle \Bigr]\\
&&+{\mathbb{E}}_{\mu}\Bigl[\bigl\langle {\mathbb{E}}_{\mu}[X_{1}|X^{n}_{2}], h_{1}\bigr\rangle\bigl\langle {\mathbb{E}}_{\mu}[X_{1}|X^{n}_{2}] ,  h_{2}\bigr\rangle \Bigr]-{\mathbb{E}}_{\mu}\Bigl[\bigl\langle {\mathbb{E}}_{\mu}[X_{1}|X_{2}], h_{1}\bigr\rangle\bigl\langle {\mathbb{E}}_{\mu}[X_{1}|X_{2}] ,  h_{2}\bigr\rangle \Bigr]\end{aligned}$$ where the last term can be decomposed as $$\begin{aligned}
&&{\mathbb{E}}_{\mu}\Bigl[\bigl\langle {\mathbb{E}}_{\mu}[X_{1}|X^{n}_{2}], h_{1}\bigr\rangle\bigl\langle {\mathbb{E}}_{\mu}[X_{1}|X^{n}_{2}] ,  h_{2}\bigr\rangle \Bigr]-{\mathbb{E}}_{\mu}\Bigl[\bigl\langle {\mathbb{E}}_{\mu}[X_{1}|X_{2}], h_{1}\bigr\rangle\bigl\langle {\mathbb{E}}_{\mu}[X_{1}|X_{2}] ,  h_{2}\bigr\rangle \Bigr]\\
=&& {\mathbb{E}}_{\mu}\Bigl[\bigl\langle {\mathbb{E}}_{\mu}[X_{1}|X^{n}_{2}]-{\mathbb{E}}_{\mu}[X_{1}|X_{2}], h_{1}\bigr\rangle\bigl\langle {\mathbb{E}}_{\mu}[X_{1}|X^{n}_{2}] ,  h_{2}\bigr\rangle \Bigr]\\
 &+& {\mathbb{E}}_{\mu}\Bigl[\bigl\langle {\mathbb{E}}_{\mu}[X_{1}|X_{2}], h_{1}\bigr\rangle\bigl\langle {\mathbb{E}}_{\mu}[X_{1}|X^{n}_{2}]-{\mathbb{E}}_{\mu}[X_{1}|X_{2}] ,  h_{2}\bigr\rangle \Bigr]\, .\end{aligned}$$ Then since conditional expectation is a contraction on $L_{2}(H,\mu,\mathcal{B})$ it follows that ${\Vert {\mathbb{E}}_{\mu}[X_{1}|X_{2}] \Vert}_{L_{2}(H,\mu,\mathcal{B})} \leq {\Vert X_{1} \Vert}_{L_{2}(H,\mu,\mathcal{B})}$ and ${\Vert {\mathbb{E}}_{\mu}[X_{1}|X^{n}_{2}] \Vert}_{L_{2}(H,\mu,\mathcal{B})} \leq {\Vert X_{1} \Vert}_{L_{2}(H,\mu,\mathcal{B})}$ for all $n$. Moreover, since ${\mathbb{E}}_{\mu}\bigl[X_{1}|X^{n}_{2}\bigr]$ converges to ${\mathbb{E}}_{\mu}\bigl[X_{1}|X_{2}\bigr]$ in $L^{2}(H,\mu,\mathcal{B})$ it follows, see e.g. [@AliprantisBorder:2006 Lem. 11.45], that $\langle {\mathbb{E}}_{\mu}\bigl[X_{1}|X^{n}_{2}\bigr],h\rangle$ converges to $\langle {\mathbb{E}}_{\mu}\bigl[X_{1}|X_{2}\bigr], h\rangle$ in $L^{2}({\mathbb{R}},\mu,\mathcal{B})$ for all $h \in H$. Therefore, the Cauchy-Schwartz inequality applied four times in the above decomposition implies that $$\lim_{n\rightarrow \infty}\langle C_{\mu_{n}}(X|X_{2})h_{1},h_{2}\rangle
=\langle C_{\mu}(X_{1}|X_{2})h_{1},h_{2}\rangle, \quad h_{1}, h_{2} \in H$$ so that we obtain $$C_{\mu}(X|X_{2})= \omega\mbox{-}\\lim_{n\rightarrow\infty}{C_{\mu_{n}}(X|X_{2})}\, .$$ Since $C^{n}$ is compatible with $H_{2}$ for all $n$, and the compatible case demonstrated in that $$\label{id_cov2}
C_{\mu_{n}}(X|X_{2})=\mathcal{H}_{1}(C^{n})$$ for all $n$, and Theorem \[thm\_short\] asserts that $$\mathcal{H}_{1}(C)= \omega\mbox{-}\\lim_{n\rightarrow\infty}{\mathcal{H}_{1}(C^{n})}\, ,$$ we conclude that $C_{\mu}(X|X_{2})=\mathcal{H}_{1}(C)$, establishing the assertion regarding the covariance operators.

For the means, observe that since $\mu$ is a probability measure, it follows that $X$ and therefore $X_{1}$ lie in the Lebesgue-Bochner space $L^{1}(H,\mu,\mathcal{B})$, and since by Diestel and Uhl [@DiestelUhl Thm. V.1.4] the conditional expectation operators are also contractions on $L^{1}(H,\mu,\mathcal{B})$ it also follows that ${\mathbb{E}}_{\mu_{n}}\bigl[X_{1}|X_{2}\bigr]$ converges to ${\mathbb{E}}_{\mu}\bigl[X_{1}|X_{2}\bigr]$ in $L^{1}(H,\mu,\mathcal{B})$. Therefore, Diestel and Uhl [@DiestelUhl Thm. V.2.8] implies that ${\mathbb{E}}_{\mu_{n}}\bigl[X_{1}|X_{2}\bigr]$ converges to $E_{\mu}\bigl[X_{1}|X_{2}\bigr]$ a.e.-$\mu$. Let the conditional means ${\mathbb{E}}_{\mu}\bigl[X|X_{2}\bigr]$ be denoted by ${\mathbb{E}}_{\mu}\bigl[X|X_{2}\bigr]=m_{t}, t\in H_{2}$. Then, since $$P^{n}m =\begin{pmatrix}
  m_{1} \\
P_{H^{n}_{2}}m_{2}
 \end{pmatrix}\, ,$$ is the mean of the measure $\mu_{n}$, the assertion in the compatible case demonstrated that the conditional means ${\mathbb{E}}_{\mu_{n}}\bigl[X|X_{2}]=m^{n}_{t}, t\in H_{2}$ are $$m^{n}_{t}= \begin{pmatrix}
  m_{1}+\hat{Q}_{n}^{*}(t-P_{H^{n}_{2}}m_{2}) \\
t
 \end{pmatrix}\, .$$ Since the conditional means ${\mathbb{E}}_{\mu_{n}}\bigl[X_{1}|X_{2}\bigr]$ converge to the conditional means ${\mathbb{E}}_{\mu}\bigl[X_{1}|X_{2}\bigr]$ a.e.-$\mu$ amounts to $m^{n}_{t} \rightarrow m_{t}$ for $\mu$-almost every $t$, the first assertion regarding the means is also proved. Now suppose that $Q_{n}$ eventually becomes the special element $Q_{n}=Q_{C^{n},H_{2}}$ defined near . Then, by definition, $R(\hat{Q}_{n})\subset {\mkern 1.5mu\overline{\mkern-1.5muR(C^{n}_{22})\mkern-1.5mu}\mkern 1.5mu}$ so that $ker(\hat{Q}^{*}_{n})\supset R(C^{n}_{22})^{\perp}$, but since $C^{n}_{22}=\Pi_{2}C^{n}\Pi_{2}^{*}=\Pi_{2}P^{n}CP^{n}\Pi_{2}^{*}=\Pi_{2}P_{H^{n}_{2}}CP_{H^{n}_{2}}\Pi_{2}^{*}$, it follows that $R(C^{n}_{22})\subset H_{2}^{n}$ and therefore $R(C^{n}_{22})^{\perp}\supset (H^{n}_{2})^{\perp}$ so that $ker(\hat{Q}^{*}_{n}) \supset (H^{n}_{2})^{\perp}$. Therefore $\hat{Q}^{*}_{n}P_{H^{n}_{2}}=\hat{Q}^{*}_{n}$, so that the final assertion follows from the previous.

Proof of Corollary \[cor\_short\]
---------------------------------

By Mourier’s Theorem, there exists a Gaussian measure $\mu$ on $H$ with mean $0$ and covariance operator $C:=A$. Looking at the end of the proof of Theorem \[thm\_main\], since conditional expectation is a contraction on $L_{2}(H,\mu,\mathcal{B})$ it follows that ${\Vert {\mathbb{E}}_{\mu}[X_{1}|X_{2}] \Vert}_{L_{2}(H,\mu,\mathcal{B})} \leq {\Vert X_{1} \Vert}_{L_{2}(H,\mu,\mathcal{B})}$ and ${\Vert {\mathbb{E}}_{\mu}[X_{1}|X^{n}_{2}] \Vert}_{L_{2}(H,\mu,\mathcal{B})} \leq {\Vert X_{1} \Vert}_{L_{2}(H,\mu,\mathcal{B})}$ for all $n$. Therefore, for $h \in H$, it follows from the Cauchy-Schwartz inequality that ${\Vert \langle {\mathbb{E}}_{\mu}[X_{1}|X^{n}_{2}], h\rangle \Vert}_{L_{2}(\mathbb{R},\mu,\mathcal{B})} \leq {\Vert X_{1} \Vert}_{L_{2}(H,\mu,\mathcal{B})}$ and ${\Vert \langle {\mathbb{E}}_{\mu}[X_{1}|X_{2}], h\rangle \Vert}_{L_{2}(\mathbb{R},\mu,\mathcal{B})} \leq {\Vert X_{1} \Vert}_{L_{2}(H,\mu,\mathcal{B})}$ for all $n$, uniformly for $h \in H$ with ${\Vert h \Vert}_{H}\leq 1$. Therefore, the Cauchy-Schwartz inequality applied four times in the decomposition at the end of the proof of Theorem \[thm\_main\] implies that $$\lim_{n\rightarrow \infty}\langle C_{\mu_{n}}(X|X_{2})h_{1},h_{2}\rangle
=\langle C_{\mu}(X_{1}|X_{2})h_{1},h_{2}\rangle, \quad h_{1}, h_{2} \in H$$ uniformly for $h_{1},h_{2} \in H$ with ${\Vert h_{1} \Vert}_{H}\leq 1$ and ${\Vert h_{2} \Vert}_{H}\leq 1$. Therefore, it follows from Halmos [@halmos1982hilbert Prob. 107] that the sequence of covariance operators converges $$C_{\mu_{n}}(X|X_{2}) \rightarrow
C_{\mu}(X|X_{2})$$ in the uniform operator topology.

According to Maniglia and Rhandi [@maniglia2004gaussian Ch. 1, Lem. 1.1.4] or Da Prato and Zabczyk [@da2014stochastic Prop. 2.16], for a Gaussian measure $\mu$ with mean $0$ and covariance operator $C$, we have $$\label{id_trace}tr(C)={\mathbb{E}}_{\mu}{{\Vert X \Vert}^{2}}\, .$$ From , by shifting to the center, we obtain that $$\label{id_cov33a}
tr\bigl(C_{\mu_{n}}(X|X_{2})\bigr)={\mathbb{E}}_{\mu}\Bigl[{\Vert X_{1}-{\mathbb{E}}_{\mu}[X_{1}|X^{n}_{2}] \Vert}^{2}
\Bigr]\,$$ and $$\begin{aligned}
tr\bigl(C_{\mu}(X|X_{2})\bigr)=
{\mathbb{E}}_{\mu}\Bigl[{\Vert X_{1}-{\mathbb{E}}_{\mu}[X_{1}|X_{2}] \Vert}^{2} \Bigr]\, ,\end{aligned}$$ and therefore the difference is $$\begin{aligned}
&&tr\bigl(C_{\mu_{n}}(X|X_{2})\bigr)-tr\bigl(C_{\mu}(X|X_{2})\bigr) \\
&=& {\mathbb{E}}_{\mu}\Bigl[{\Vert X_{1}-{\mathbb{E}}_{\mu}[X_{1}|X^{n}_{2}] \Vert}^{2}
\Bigr]-E_{\mu}\Bigl[{\Vert X_{1}-{\mathbb{E}}_{\mu}[X_{1}|X_{2}] \Vert}^{2} \Bigr]\\
&=& {\mathbb{E}}_{\mu}\Bigl[\bigl\langle {\mathbb{E}}_{\mu}[X_{1}|X^{n}_{2}] -{\mathbb{E}}_{\mu}[X_{1}|X_{2}],\,
{\mathbb{E}}_{\mu}[X_{1}|X^{n}_{2}] +{\mathbb{E}}_{\mu}[X_{1}|X_{2}]-2X_{1}\bigr\rangle \Bigr]\,.\end{aligned}$$ Therefore, the Cauchy-Schwartz inequality, the $L^{2}$ convergence of ${\mathbb{E}}_{\mu}[X_{1}|X^{n}_{2}]$ to $ {\mathbb{E}}_{\mu}[X_{1}|X_{2}]$, and the uniform $L^{2}$ boundedness of ${\mathbb{E}}_{\mu}[X_{1}|X^{n}_{2}]$, $ {\mathbb{E}}_{\mu}[X_{1}|X_{2}]$ and $X_{1}$, implies that $$\lim_{n\rightarrow \infty}{tr\bigl(C_{\mu_{n}}(X|X_{2})\bigr)}=tr\bigl(C_{\mu}(X|X_{2})\bigr)\, .$$ Since $C_{\mu_{n}}(X|X_{2}) \rightarrow C_{\mu}(X|X_{2})$ in the uniform operator topology, it follows from Kubrusly [@Kubrusly], see [@kubrusly1985convergence Thm. 2 & Sec. 4], that $C_{\mu_{n}}(X|X_{2}) \rightarrow C_{\mu}(X|X_{2})$ in the trace norm topology. Since asserts that $C_{\mu_{n}}(X|X_{2})=\mathcal{H}_{1}(C^{n})$ and Theorem \[thm\_main\] asserts that $C_{\mu}(X|X_{2})=\mathcal{H}_{1}(C)$, the identification $A:=C$ completes the proof.

Acknowledgments {#acknowledgments .unnumbered}
===============

The authors gratefully acknowledge this work supported by the Air Force Office of Scientific Research under Award Number FA9550-12-1-0389 (Scientific Computation of Optimal Statistical Estimators).
