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:
[egin{cases}
(Af)(x)=if'(x),quad forall fin D(A),\
D(A)={f,f'in L^2(0,1)|f(0)=f(1)=0}.
end{cases}]
证明:
egin{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 angle=
egin{cases}
1, & n=m,\
0, & n eq m,
end{cases}forall n,mgeq 1.]
且对任意的$xin H$,存在复数序列${a_n}_1^infty$使得$displaystyle x=sum_{n=1}^infty a_ne_n$.
egin{enumerate}
item 记$displaystyleell^2=left{{a_n}_1^inftyleft|sum_{n=1}^{infty}|a_n|^2<infty ight. ight}$.试证对任意的$xin H$,存在${a_n} _1^inftyinell^2$,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 ngeq 1.]
end{enumerate}

item 设$T$为Banach空间$X$上的线性有界算子且对任意的$xin X$, $displaystylelim_{n oinfty}left|T^nx ight|=0$.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.
egin{enumerate}
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$displaystylelim_{n oinfty}xi_k^{(n)}=xi_k^{(0)},forall kgeq 1$.

item 如果$T$满足$displaystylesum_{k=1}^{infty}left|Te_k ight|^2<infty$,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)
end{enumerate}

item 试求$L^2(0,1)$中以
[k(t,s)=egin{cases}
s(1-t),&0leq sleq t,\
t(1-s),&t < sleq 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$Kvarphi=lambdavarphi$化成等价的关于$varphi$的微分方程边值问题)
end{enumerate}

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+frac12sin f(x).]

item 设复平面上的序列${lambda_n}_{n=1}^infty$满足$sup_n |mathrm{Im}lambda_n|<infty,inf_{n eq m,n,mgeq1}|lambda_n-lambda_m|>0$ (这样的序列称为可分离的).定义
[n^+(r)=sup_{xinmathbb{R}}{mathrm{Re}lambda_nin[x,x+r) ext{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,tinmathbb{R}]
由此证明:[D(Lambda)=lim_{r o+infty}frac{n^+(r)}{r}]存在且$D(Lambda)<+infty$.

item 证明所有$n imes n$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 ext{的最大特征值,quad $forall Ain H_n$}]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$ar{lambda}$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, ${U_n,ngeq 1}$ 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 ${U_n,ngeq 1}$. 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 ight|V_1=t ight] =left( mathrm{E}left[ U_1 ight] ight) imes left[ 2+mleft( t ight) ight].
]
Thus
egin{align*}
mathrm{E}left[ V_1+C_1 ight] &=int_0^{infty}{mu e^{-mu t}left( mathrm{E}left[ U_1 ight] ight) imes left[ 2+mleft( t ight) ight] dt}
\
&=2mathrm{E}left[ U_1 ight] +mathrm{E}left[ U_1 ight] int_0^{infty}{mu e^{-mu t}mleft( t ight) dt}
\
&=2mathrm{E}left[ U_1 ight] +mathrm{E}left[ U_1 ight] imes frac{widetilde{F}left( mu ight)}{1-widetilde{F}left( mu ight)}=frac{2-widetilde{F}left( mu ight)}{1-widetilde{F}left( mu ight)}mathrm{E}left[ U_1 ight].
end{align*}
It follows from Lecture 2, Theorem 6.3 that
[
lim_{t ightarrow infty}Pleft( Qleft( t ight) =2 ight) =frac{mathrm{E}left[ V_1 ight]}{mathrm{E}left[ V_1+C_1 ight]}=frac{1}{mu imes mathrm{E}left[ U_1 ight]}cdot frac{1-widetilde{F}left( mu ight)}{2-widetilde{F}left( mu ight)}.
]

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 ight|W_1=t ight] =left( Eleft[ U_1 ight] ight) imes left[ 2+mleft( t ight) ight].
]
Thus
egin{align*}
mathrm{E}left[ W_1+D_1 ight] &=int_{-infty}^{infty}{left( mathrm{E}left[ U_1 ight] ight) imes left[ 2+mleft( t ight) ight] dFleft( t ight)}
\
&=2mathrm{E}left[ U_1 ight] +mathrm{E}left[ U_1 ight] int_{-infty}^{infty}{mleft( t ight) dFleft( t ight)}.
end{align*}
It follows from Lecture 2, Theorem 6.3 that
[
lim_{t ightarrow infty}Pleft( Qleft( t ight) =1 ight) =frac{mathrm{E}left[ W_1 ight]}{mathrm{E}left[ W_1+D_1 ight]}=frac{1}{mathrm{E}left[ U_1 ight]}cdot frac{int_{-infty}^{infty}{xdFleft( t ight)}}{2+int_{-infty}^{infty}{mleft( t ight) dFleft( t ight)}}.
]

考察取值在非负整数集$E$上的随机过程$X={X_t,tin T=[0,+infty)}$,如果对一切$T$中的时刻$0leq t_1<t_2<cdots<t_{n+1}$及满足$P(X_{t_k}=i_k,1leq kleq n)>0$的任意状态$i_kin E (1leq kleq n)$成立着
[P{X_{t_{n+1}}=j|X_{t_k}=i_k,1leq kleq 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.