英国代考|7CCMCS04 Stochastic Processes and Applications随机过程和应用

For $\eta \in k_{0}$ and any Hilbert space $\mathcal{H}$, we introduce bounded linear map $C_{\eta}: \mathcal{H} \otimes k_{0} \rightarrow \mathcal{H}$ given by $C_{\eta}(u \otimes \xi):=u\langle\eta, \xi\rangle$. Thus, $\langle\eta, R\rangle v=C_{\eta}(R v)$. We claim that for $\xi \in \operatorname{Dom}\left(r_{t} \otimes I\right), C_{\eta}(\xi) \in \operatorname{Dom}\left(r_{t}\right)$ and
$$C_{\eta}\left(r_{t} \otimes I\right) \xi=r_{t} C_{\eta} \xi$$

Clearly, the above equality holds for $\xi \in \mathcal{D}{1} \otimes{\text {alg }} k_{0}$. Using the fact that $\mathcal{D}{1} \otimes{\text {alg }}$ $k_{0}$ is a core for $r_{t} \otimes I$, given $\xi \in \operatorname{Dom}\left(r_{t} \otimes I\right)$, we choose sequence $\xi_{n}$ from $\mathcal{D}{1} \otimes{\text {alg }} k_{0}$ such that $\xi_{n} \rightarrow \xi,\left(r_{t} \otimes I\right) \xi_{n} \rightarrow\left(r_{t} \otimes I\right) \xi$. Hence we have $C_{n} \xi_{n} \rightarrow$ $C_{\eta} \xi, r_{t} C_{\eta} \xi_{n}=C_{\eta}\left(r_{t} \otimes I\right) \xi_{n} \rightarrow C_{\eta}\left(r_{t} \otimes I\right) \xi^{\xi}$. Since $r_{t}$ is closed, the claim is proved. By a similar argument using the estimate (5.8), we conclude that $C_{\eta} \xi \in \operatorname{Dom}\left(M_{I}\right)$ with
$$C_{\eta} \tilde{M}{r} \xi=M{t} C_{\eta} \xi .$$
Furthermore,
$$\left\langle H_{t} P(w e(g)), P\left(w^{\prime} e(f)\right)\right\rangle=\left\langle\tilde{M}_{t} w, w^{\prime}\right\rangle$$

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