Riemann integral of operator valued function on Hilbert Space

Question

Let $\mathcal{H}$ be a Hilbert space and $F:[a,b]\to B({\mathcal{H}})$ be a continuous function. Suppose $P$ is a spectral measure on $\mathcal{B}([a,b])$. Suppose $F(t)P(\Delta)=P(\Delta)F(t)$ for all $t\in [a,b]$ and $\Delta\in\mathcal{B}([a,b])$. Let $S=\{a=t_1<t_2,\cdots, t_n=b\}$ be a partition of $[a,b]$. Show that the following limit $$\lim_{||S||\to 0}\left[\sum_{i=1}^nF(t_i)P([t_i,t_{i+1}))+F(b)P(\{b\})\right]v$$ exists for all $v\in\mathcal{H}$ where $||S||$ is the norm of the partition $S$.

I tried it in the following way: Take a sequence of partition $S_n$ s.t $||S_n||$ goes to $0$. Let us denote the above expression by $I(S_n)$. Then I tried to argue that $I(S_n)$ is Cauchy. But I am stuck at this stage.

Answer 1

Let $v\in\mathcal{H}$ be given. Without loss of generality, assume $v$ is a unit vector. Let $\epsilon > 0$ be given. Then there exists $\delta > 0$ such that $$ \|F(t)-F(t')\| < \epsilon \mbox{ whenever } |t-t'| < \delta. $$ Let $\mathcal{P}$ be a partition described as disjoint intervals $I_n=[t_{n-1},t_{n})$ with union is $[a,b)$. Similarly let $\mathcal{P}'$ be another partition described in terms of intervals $I_n'=[t_{n-1}',t_{n}')$. Let $\mathcal{P}''$ be a common refinement of the two partitions so that it's intervals $I_{n}''$ are subintervals of some $I_k$ and some $I_l'$. Then, for any unit vector $v\in\mathcal{H}$, $$ \|\sum_{\mathcal{P}}F(t_n)P(I_n)v-\sum_{\mathcal{P}''}F(t_n'')P(I_n'')v\|\\ = \|\sum_{n}F(t_n)P(I_n)v-\sum_{n}\sum_{\{ n'' : I_n''\subset I_n\}}F(t_n'')P(I_n'')v\| \\ = \|\sum_{n}\sum_{\{ n'' : I_n''\subset I_n\}}(F(t_n)-F(t_n''))P(I_n'')v\| \\ = \|\sum_{n}\sum_{\{ n'' : I_n''\subset I_n\}}P(I_n'')(F(t_n)-F(t_n''))P(I_n'')v\| \\ = \left(\sum_{n}\sum_{\{ n'' : I_n''\subset I_n\}}\|P(I_n'')(F(t_n)-F(t_n''))P(I_n'')v\|^2\right)^{1/2} \\ = \left(\sum_{n}\sum_{\{ n'' : I_n''\subset I_n\}}\|(F(t_n)-F(t_n''))P(I_n'')v\|^2\right)^{1/2} \\ \le \epsilon\left(\sum_{n}\sum_{\{ n'' : I_n''\subset I_n\}}\|(P(I_n'')v\|^2\right)^{1/2} \\ = \epsilon\|P[a,b)v\| \le \epsilon\|v\| = \epsilon. $$ So, all partitions whose norms are bounded by $\delta$ are within $\epsilon$ of each other. Because $\epsilon > 0$ was arbitrary, this means that the Cauchy criteria for the convergence of the integral holds.