题目

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\item 在Hilbert空间$H=L^2(0,1)$中定义线性算子$A$如下:
$$\begin{cases} (Af)(x)=if'(x),\quad \forall f\in D(A),\\ D(A)=\{f,f'\in L^2(0,1)|f(0)=f(1)=0\}. \end{cases}$$
证明:
\begin{enumerate}
\item $A$是$H$上的对称算子;

\item 试求$A$的伴随算子$A^\ast$,包括$A^\ast$的表达式和定义域$D(A^\ast)$.
\end{enumerate}

\item 可分复Hilbert空间$H$上的基$\{e_n\}_1^\infty$称为规范直交的,是指它满足
$$\langle e_n,e_m\rangle= \begin{cases} 1, & n=m,\\ 0, & n\neq m, \end{cases}\forall n,m\geq 1.$$
且对任意的$x\in H$,存在复数序列$\{a_n\}_1^\infty$使得$\displaystyle x=\sum_{n=1}^\infty a_ne_n$.
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\item 记$\displaystyle\ell^2=\left\{\{a_n\}_1^\infty\left|\sum_{n=1}^{\infty}|a_n|^2<\infty\right.\right\}$.试证对任意的$x\in H$,存在$\{a_n\} _1^\infty\in\ell^2$,使得
$$x=\sum_{n=1}^\infty a_ne_n.$$

\item 试证: $H$中所有规范直交基都是等价的:即对$H$中任意两个规范直交基$\{e_n\}_1^\infty$和$\{f_n\}_1^\infty$,必存在$H$上的有界可逆算子$T$,使$T^{-1}$有界,且
$$Te_n=f_n,\qquad \forall n\geq 1.$$
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\item 设$T$为Banach空间$X$上的线性有界算子且对任意的$x\in X$, $\displaystyle\lim_{n\to\infty}\left\|T^nx\right\|=0$.证明: $$(\lambda I-T)^{-1}=\sum_{n=1}^{\infty}\frac{T^n}{\lambda^{n+1}},\quad \forall |\lambda|>1.$$

\item 设$\{e_n\}$可分为Hilbert空间$X$中的规范直交基, $T$为$X$中有界线性算子.
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\item 试证$X$中弱收敛等价于"按坐标收敛",即$\displaystyle x_n=\sum_{k=1}^{\infty}\xi_k^{(n)}e_k$弱收敛于元$\displaystyle x_0=\sum_{k=1}^{\infty}\xi_k^{(0)}e_k$,当且仅当$\displaystyle\lim_{n\to\infty}\xi_k^{(n)}=\xi_k^{(0)},\forall k\geq 1$.

\item 如果$T$满足$\displaystyle\sum_{k=1}^{\infty}\left\|Te_k\right\|^2<\infty$,试证$T$是紧算子. ($T$称为紧算子,是指$T$把$X$中任意弱收敛序列变成强收敛序列)
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\item 试求$L^2(0,1)$中以
$$k(t,s)=\begin{cases} s(1-t),&0\leq s\leq t,\\ t(1-s),&t < s\leq 1 \end{cases}$$
为积分核的积分算子$K$:
$$(Kf)(t)=\int_{0}^{1}k(t,s)f(s)ds$$
的本征值和本征函数. (提示:设法把本征值问题$K\varphi=\lambda\varphi$化成等价的关于$\varphi$的微分方程边值问题)
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3小时完成,每题20分,满分100分.

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\item 证明在$[-1,1]$上存在唯一的连续函数$f$,使得
$$f(x)=x+\frac12\sin f(x).$$

\item 设复平面上的序列$\{\lambda_n\}_{n=1}^\infty$满足$\sup_n |\mathrm{Im}\lambda_n|<\infty,\inf_{n\neq m,n,m\geq1}|\lambda_n-\lambda_m|>0$ (这样的序列称为可分离的).定义
$$n^+(r)=\sup_{x\in\mathbb{R}}\{\mathrm{Re}\lambda_n\in[x,x+r)\text{的$\lambda_n$的个数}\}.$$ 这里$\mathrm{Im}$表示复数的复部, $\mathrm{Re}$表示复数的实部, $\mathbb{R}$表示实直线, $|\cdot|$表示复数间的距离.证明$n^+(r)$是次可加的:
$$n^+ (s+t)\leq n^+(s)+n^+(t),\quad \forall s,t\in\mathbb{R}$$
由此证明: $$D(\Lambda)=\lim_{r\to+\infty}\frac{n^+(r)}{r}$$ 存在且$D(\Lambda)<+\infty$.

\item 证明所有$n\times n$复矩阵在通常的加法,数量乘法形成的线性空间$H_n$在如下范数
$$\|A\|^2=AA^\ast\text{的最大特征值,\quad $\forall A\in H_n$}$$ 下形成Banach空间.这里$A^\ast$表示矩阵$A$的共轭转置.

\item 设$T$为Hilbert空间$H$上的线性有界算子.证明: (a) 如果$\|T\|\leq1$则$T$与其共轭算子$T^\ast$有相同的不动点: $Tx=x$当且仅当$T^\ast x=x$.

(b)如果$\lambda$是$T$的本征值,问$\lambda$的共轭$\bar{\lambda}$是否一定为$T^\ast$的本征值?是否一定为$T^\ast$的谱点?

\item 证明Banach空间$X$上不存在这样的线性有界算子$A,B$使得
$$AB-BA=I,$$ 其中$I$为$X$上的单位算子. (提示:否则对任意正整数$n$, $(n+1)B^n=AB^{n+1}-B^{n+1}A$).
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Let $U_n$ be the interarrival time between the $n-1$st and the $n$th customers, $\{U_n,n\geq 1\}$ is a sequence of i.i.d. random variables. Let $Q(t)$ be the number of customers in the system at time $t$, the process $Q(t)$ takes three values $0,1$ and $2$. The $n$th time $V_n$ when $Q(t)$ takes $2$ is exponential with parameter $\mu$. Let $C_n$ be the $n$th time when $Q(t)$ takes $0$ or $1$, $W_n$ be the $n$th time when $Q(t)$ takes $1$, $D_n$ be the $n$th time when $Q(t)$ takes $2$ or $0$, and $N(t)$ the number of potential arrivals during $(0,t]$. Then $N(t)$ is the renewal process generated by $\{U_n,n\geq 1\}$. Give $V_1=t$,then
$$V_1+C_1=S_{N(t)+2}.$$
By Wald' equation we have
$$\mathrm{E}\left[ \left. V_1+C_1 \right|V_1=t \right] =\left( \mathrm{E}\left[ U_1 \right] \right) \times \left[ 2+m\left( t \right) \right].$$
Thus
$$\begin{align*} \mathrm{E}\left[ V_1+C_1 \right] &=\int_0^{\infty}{\mu e^{-\mu t}\left( \mathrm{E}\left[ U_1 \right] \right) \times \left[ 2+m\left( t \right) \right] dt} \\ &=2\mathrm{E}\left[ U_1 \right] +\mathrm{E}\left[ U_1 \right] \int_0^{\infty}{\mu e^{-\mu t}m\left( t \right) dt} \\ &=2\mathrm{E}\left[ U_1 \right] +\mathrm{E}\left[ U_1 \right] \times \frac{\widetilde{F}\left( \mu \right)}{1-\widetilde{F}\left( \mu \right)}=\frac{2-\widetilde{F}\left( \mu \right)}{1-\widetilde{F}\left( \mu \right)}\mathrm{E}\left[ U_1 \right]. \end{align*}$$
It follows from Lecture 2, Theorem 6.3 that
$$\lim_{t\rightarrow \infty}P\left( Q\left( t \right) =2 \right) =\frac{\mathrm{E}\left[ V_1 \right]}{\mathrm{E}\left[ V_1+C_1 \right]}=\frac{1}{\mu \times \mathrm{E}\left[ U_1 \right]}\cdot \frac{1-\widetilde{F}\left( \mu \right)}{2-\widetilde{F}\left( \mu \right)}.$$

Give $W_1=t$,then
$$W_1+D_1 =S_{N(t)+2}.$$
By Wald' equation we have
$$\mathrm{E}\left[ \left. W_1+D_1 \right|W_1=t \right] =\left( E\left[ U_1 \right] \right) \times \left[ 2+m\left( t \right) \right].$$
Thus
$$\begin{align*} \mathrm{E}\left[ W_1+D_1 \right] &=\int_{-\infty}^{\infty}{\left( \mathrm{E}\left[ U_1 \right] \right) \times \left[ 2+m\left( t \right) \right] dF\left( t \right)} \\ &=2\mathrm{E}\left[ U_1 \right] +\mathrm{E}\left[ U_1 \right] \int_{-\infty}^{\infty}{m\left( t \right) dF\left( t \right)}. \end{align*}$$
It follows from Lecture 2, Theorem 6.3 that
$$\lim_{t\rightarrow \infty}P\left( Q\left( t \right) =1 \right) =\frac{\mathrm{E}\left[ W_1 \right]}{\mathrm{E}\left[ W_1+D_1 \right]}=\frac{1}{\mathrm{E}\left[ U_1 \right]}\cdot \frac{\int_{-\infty}^{\infty}{xdF\left( t \right)}}{2+\int_{-\infty}^{\infty}{m\left( t \right) dF\left( t \right)}}.$$

考察取值在非负整数集$E$上的随机过程$X=\{X_t,t\in T=[0,+\infty)\}$,如果对一切$T$中的时刻$0\leq t_1<t_2<\cdots<t_{n+1}$及满足$P(X_{t_k}=i_k,1\leq k\leq n)>0$的任意状态$i_k\in E (1\leq k\leq n)$成立着
$$P\{X_{t_{n+1}}=j|X_{t_k}=i_k,1\leq k\leq n\}=P\{X_{t_{n+1}}=j|X_{t_n}=i_n\},$$
则称$X$是连续时间马尔可夫链.