本节讨论算子 $I_{g,\varphi}: H(p,\,q,\,\phi) \rightarrow {\cal Z}_0$ 的有界性和紧性.

$\bf 定理 3.1$ 设 $\varphi$ 是 $D\rightarrow D$ 的一个解析自映射,$\phi$ 是正规函数, $0<p,\,q<\infty$ 和 $g\in H(D)$. 则 $I_{g,\varphi}$: $H(p,\,q,\,\phi)\rightarrow {\cal Z}_{0}$ 是有界算子的充分必要条件是 $I_{g,\varphi}$: $H(p,\,q,\,\phi)\rightarrow {\cal Z}$ 是有界算子且

$$\begin{equation} \label{e20} \lim_{|z|\rightarrow1}(1-|z|^{2})|(\varphi'(z))^{2}g'(\varphi(z))+\varphi''(z)g(\varphi(z))|=0,\end{equation}$$

(3.1)

$$\begin{equation} \label{e21} \lim_{|z|\rightarrow1}(1-|z|^{2})|(\varphi'(z))^{2}g(\varphi(z))|=0. \end{equation}$$

(3.2)

$\bf 证$ 必要性. 设 $I_{g,\varphi}$: $H(p,\,q,\, \phi)\rightarrow {\cal Z}_{0}$是有界算子,则对任意的 $f\in H(p,\,q,\,\phi)$, $I_{g,\varphi}f\in {\cal Z}_{0}$. 取检验函数$f(z)=z \in H(p,\,q,\,\phi)$,得到 $$\lim_{|z|\rightarrow 1}(1-|z|^{2})|(\varphi'(z))^{2}g'(\varphi(z))+\varphi''(z)g(\varphi(z))|=0.$$ 即(3.1)式成立.

取检验函数 $f(z)=z^2\in H(p,\,q,\,\phi)$, 由三角不等式及 $\|\varphi\|_{\infty}\leq 1$,得到

$$\begin{eqnarray} &&2(1-|z|^{2})|\varphi'(z)|^2|g(\varphi(z))|\nonumber\\ &\leq &(1-|z|^{2})|(I_{g,\varphi}f)''(z)|+2(1-|z|^{2})|\varphi(z)||(\varphi'(z))^{2}g'(\varphi(z))+\varphi''(z)g(\varphi(z))|\nonumber\\ &\leq &(1-|z|^{2})|(I_{g,\varphi}f)''(z)|+2(1-|z|^{2})|(\varphi'(z))^{2}g'(\varphi(z))+\varphi''(z)g(\varphi(z))|, \end{eqnarray}$$

(3.3)

由此及 (3.1)式,我们得到 $$\lim_{|z|\rightarrow1}(1-|z|^{2})|\varphi'(z)|^2\cdot|g(\varphi(z))|=0.$$ 即 (3.2)式成立.

充分性. 对任意的多项式 $p$,由于 $$\begin{eqnarray*} \label{x1} &&(1-|z|^{2})|(I_{g,\varphi}p)''(z)|\\ &\leq& (1-|z|^{2})|p''(\varphi(z))||\varphi'(z)|^2\cdot|g(\varphi(z))|\\ &&+(1-|z|^{2})|p'(\varphi(z))|\cdot|(\varphi'(z))^{2}g'(\varphi(z))+\varphi''(z)g(\varphi(z))|\\ &\leq& \|p''\|_\infty(1-|z|^{2})|\varphi'(z)|^2|g(\varphi(z))|\\ &&+\|p'\|_\infty(1-|z|^{2})\cdot|(\varphi'(z))^{2}g'(\varphi(z))+\varphi''(z)g(\varphi(z))|, \end{eqnarray*}$$ 利用 (3.1) 和 (3.2)式,得出 $I_{g, \varphi}p\in {\cal Z}_{0}$. 由于多项式的全体在 $H(p,\,q,\,\phi)$ 中稠密, 所以对任意的 $f\in H(p,\,q,\,\phi)$, 存在多项式序列 $p_k$,使得 $$\lim\limits_{k\rightarrow \infty}\|p_k-f\|_{p,\,q,\,\phi}=0.$$ $I_{g,\varphi}: H(p,\,q,\,\phi)\rightarrow {\cal Z}$ 的有界性给出 $$\|I_{g,\varphi}f-I_{g,\varphi}p_k\|_{{\cal Z}} \leq\|I_{g,\varphi}\|_{H(p,\,q,\,\phi)\rightarrow{\cal Z}}\|p_k-f\|_{p,\,q,\,\phi} \rightarrow 0,k\rightarrow \infty,$$ 由于 ${\cal Z}_0$ 是 ${\cal Z}$ 的闭子空间,因此 $I_{g,\varphi}(H(p,\,q,\,\phi))\subseteq{\cal Z}_0$, 于是 $I_{g,\varphi}: H(p,\,q,\,\phi)\rightarrow {\cal Z}_0$ 是有界算子.

$\bf 定理 3.2$ 设 $\varphi$ 是 $D\rightarrow D$ 的一个解析自映射,$\phi$ 是正规函数, $0<p,\,q<\infty$ 和 $g\in H(D)$. 则下列条件等价.

(a)~ $I_{g,\varphi}$: $H(p,\,q,\,\phi)\rightarrow {\cal Z}_{0}$ 是紧算子.

(b)~ 当 $\|\varphi\|_\infty <1$ 时,(3.1),(3.2) 式成立;

当 $\|\varphi\|_\infty =1$ 时,(3.1),(3.2) 式成立以及

$$\begin{equation} \label{e10}\lim_{|\varphi(z)|\rightarrow1}\frac{(1-|z|^{2})|\varphi'(z)|^2\cdot|g(\varphi(z))|}{\phi(|\varphi(z)|)\left(1-|\varphi(z)|^{2}\right)^{2+\frac{1}{q}} }=0 , \end{equation}$$

(3.4)

$$\begin{equation} \label{e11}\lim_{|\varphi(z)|\rightarrow1}\frac{(1-|z|^{2})|(\varphi'(z))^{2}g'(\varphi(z))+\varphi''(z)g(\varphi(z))|}{\phi(|\varphi(z)|)\left(1-|\varphi(z)|^{2}\right)^{1+\frac{1}{q}} }=0. \end{equation}$$

(3.5)

(c)

$$\begin{equation} \label{e27} \lim_{|z|\rightarrow1}\frac{(1-|z|^{2})|\varphi'(z)|^2\cdot|g(\varphi(z))|}{\phi(|\varphi(z)|)\left(1-|\varphi(z)|^{2}\right)^{2+\frac{1}{q}} }=0, \end{equation}$$

(3.6)

$$\begin{equation} \label{e28}\lim_{|z|\rightarrow1}\frac{(1-|z|^{2})|(\varphi'(z))^{2}g'(\varphi(z))+\varphi''(z)g(\varphi(z))|}{\phi(|\varphi(z)|)\left(1-|\varphi(z)|^{2}\right)^{1+\frac{1}{q}} }=0 .\end{equation}$$

(3.7)

$\bf 证$ (b)$\Rightarrow$(c). 如果 $\|\varphi\|_\infty <1$,利用 $\phi$ 的正规性,存在数 $s>0$,使得 $\frac{\phi(r)}{(1-r)^{s}}$ 在 $[0,1)$ 上单调减少,于是 $$\phi(\|\varphi\|_\infty)\leq \frac{\phi(|\varphi(z)|)(1-\|\varphi\|_\infty)^{s}}{(1-|\varphi(z)|)^{s}}\leq \phi(|\varphi(z)|),z\in D.$$ 利用 (3.1) 和 (3.2)式,我们得到 $$\lim_{|z|\rightarrow1}\frac{(1-|z|^{2})|\varphi'(z)|^2\cdot|g(\varphi(z))|}{\phi(|\varphi(z)|)\left(1-|\varphi(z)|^{2}\right)^{2+\frac{1}{q}} } \leq \lim_{|z|\rightarrow1}\frac{(1-|z|^{2})|\varphi'(z)|^2\cdot|g(\varphi(z))|}{\phi(\|\varphi\|_\infty)\left(1-\|\varphi\|_\infty^{2}\right)^{2+\frac{1}{q}} }=0$$ 和 $$\begin{eqnarray*} &&\lim_{|z|\rightarrow1}\frac{(1-|z|^{2})|(\varphi'(z))^{2}g'(\varphi(z))+\varphi''(z)g(\varphi(z))|}{\phi(|\varphi(z)|)\left(1-|\varphi(z)|^{2}\right)^{1+\frac{1}{q}} }\nonumber\\ &\leq& \lim_{|z|\rightarrow1}\frac{(1-|z|^{2})|(\varphi'(z))^{2}g'(\varphi(z))+\varphi''(z)g(\varphi(z))|}{\phi(\|\varphi\|_\infty)\left(1-\|\varphi\|_\infty^{2}\right)^{1+\frac{1}{q}} }=0, \end{eqnarray*}$$ 由此知 (3.6) 和 (3.7)式 成立.

如果 $\|\varphi\|_\infty =1$,由(3.4)式可得, 对于任意 $\varepsilon> 0$,存在 $r\in (0,1),$使当 $r<|\varphi(z)|<1$时, 有 $$\frac{(1-|z|^{2})|\varphi'(z)|^2\cdot|g(\varphi(z))|}{\phi(|\varphi(z)|)\left(1-|\varphi(z)|^{2}\right)^{2+\frac{1}{q}} }<\varepsilon,$$ 由 (3.2)式可得,对于 $\varepsilon> 0,$存在 $\rho\in (0,1),$ 使当 $\rho<|z|<1$ 时,有

$$\begin{equation} \label{e24}(1-|z|^{2})|(\varphi'(z))^{2}g(\varphi(z))|<\varepsilon\inf_{t\in [0,r]}\phi(t)(1-t^{2})^{2+\frac{1}{q}}. \end{equation}$$

(3.8)

于是,当 $\rho<|z|<1,r<|\varphi(z)|<1$时,有

$$\begin{equation} \label{e25}\frac{(1-|z|^{2})|\varphi'(z)|^2\cdot|g(\varphi(z))|}{\phi(|\varphi(z)|)\left(1-|\varphi(z)|^{2}\right)^{2+\frac{1}{q}} }<\varepsilon. \end{equation}$$

(3.9)

另一方面,若 $\rho<|z|<1$,$|\varphi(z)|\leq r,$ 由(3.8)式得

$$\begin{equation} \label{e26}\frac{(1-|z|^{2})|\varphi'(z)|^2\cdot|g(\varphi(z))|}{\phi(|\varphi(z)|)\left(1-|\varphi(z)|^{2}\right)^{2+\frac{1}{q}} }\leq \frac{(1-|z|^{2})|\varphi'(z)|^2\cdot|g(\varphi(z))|}{\inf_{t\in [0,r]}\phi(t)(1-t^{2})^{2+\frac{1}{q}}}<\varepsilon. \end{equation}$$

(3.10)

由(3.9),(3.10)式得(3.6)式成立.

同样,由(3.5)和(3.1)式得(3.7)式成立.

(c)$\Rightarrow$(a). 由(3.6) 和 (3.7)式得 (2.1) 和 (2.2) 式成立, 从而由引理2.4知 $I_{g,\varphi}$: $H(p,\,q,\,\phi)\rightarrow {\cal Z}$ 是有界算子. 对于任意 $f\in H(p,\,q,\,\phi),$在引理2.1中分别取 $n=1$,$n=2,$ 得

$\begin{array}{l} (1 - |z{|^2})|({I_{g,\phi }}f)\prime \prime (z)|\\ \le (1 - |z{|^2})|f\prime \prime (\varphi (z))\left| {{\rm{ }}\cdot{\rm{ }}} \right|\varphi \prime (z){|^2}\cdot{\rm{ }}|g(\varphi (z))|\\ + (1 - |z{|^2})|f\prime (\varphi (z))\left| {{\rm{ }}\cdot{\rm{ }}} \right|{(\varphi \prime (z))^2}g\prime (\phi (z)){\rm{ }} + \varphi \prime \prime (z)g(\varphi (z))|\\ \le C\frac{{(1 - |z{|^2})|\varphi \prime (z){|^2}\cdot{\rm{ }}|g(\varphi (z))|}}{{\phi (|\varphi (z)|){\rm{ }}{{(1 - |\varphi (z){|^2})}^{2 + \frac{1}{q}}}}}||f|{|_{p,{\rm{ }}q,{\rm{ }}\varphi }}\\ + C\frac{{(1 - |z{|^2})|{{(\varphi \prime (z))}^2}g\prime (\varphi (z)){\rm{ }} + \varphi \prime \prime (z)g(\varphi (z))|}}{{\phi (|\varphi (z)|){\rm{ }}{{(1 - |\varphi (z){|^2})}^{1 + \frac{1}{q}}}}}||f|{|_{p,{\rm{ }}q,{\rm{ }}\varphi }} \end{array}$

(3.11)

由 (3.6),(3.7)式 和 (3.11) 式可知, 对于任意 $f\in H(p,\,q,\,\phi)$,$I_{g,\varphi}f\in {\cal Z}_{0}.$ 于是 $I_{g,\varphi}(H(p,\,q,\,\phi))\subseteq {\cal Z}_{0},$ 所以 $I_{g,\varphi}$: $H(p,\,q,\,\phi)\rightarrow {\cal Z}_{0}$ 是有界算子. 故集合 $$I_{g,\varphi}\left(\{f\in H(p,\,q,\,\phi): \|f\|_{p,\,q,\,\phi}\leq 1\}\right)$$ 在 ${\cal Z}_0$ 中是有界的,由引理2.3可知,要证 $I_{g,\varphi}$: $H(p,\,q,\,\phi)\rightarrow {\cal Z}_{0}$ 是紧算子,只要证

$$\begin{equation} \label{e30}\lim_{|z|\rightarrow1}\sup_{\|f\|_{p,\,q,\,\phi}\leq 1}(1-|z|^{2})|(I_{g,\varphi}f)''(z)|=0,\end{equation}$$

(3.12)

即可. 在不等式 (3.11) 中两边关于集合 $\{f\in H(p,\,q,\,\phi): \|f\|_{p,\,q,\,\phi}\leq 1\}$ 取上确界,然后令 $|z|\rightarrow 1$,由 (3.6),(3.7)式 可得 (3.12) 式成立,所以 $I_{g,\varphi}$: $H(p,\,q,\,\phi)\rightarrow {\cal Z}_{0}$ 是紧算子.

(a)$\Rightarrow$(b). 设 $I_{g,\varphi}$: $H(p,\,q,\,\phi)\rightarrow {\cal Z}_{0}$是紧算子,则 $I_{g,\varphi}$: $H(p,\,q,\,\phi)\rightarrow {\cal Z}_{0}$是有界算子,由定理3.1,我们得到 (3.1) 和 (3.2)式成立.

如果 $\|\varphi\|_\infty =1$,现在假设(3.4)式不成立, 则存在 $\varepsilon_{0}>0,$一序列 $\{z_k\}_{k\in N}\subset D$满足$\lim\limits_{k\rightarrow\infty}|\varphi(z_k)|=1,$及 $$\frac{(1-|z_k|^{2})|\varphi'(z_k)|^2\cdot|g(\varphi(z_k))|}{\phi(|\varphi(z_k)|)(1-|\varphi(z_k)|^{2})^{2+\frac{1}{q}}}\geq \varepsilon_{0}>0 .$$ 取文献[1,定理 2] 中检验函数列 $\{h_k\}_{k\in N}$,即 $$h_k(z)=C_{1}\frac{(1-|\varphi(z_k)|^{2})^{t+2}}{\phi(|\varphi(z_k)|)\left(1-\overline{\varphi(z_k)}z\right)^{2+t+\frac{1}{q}}}- C_{2}\frac{(1-|\varphi(z_k)|^{2})^{t+1}}{\phi(|\varphi(z_k)|)\left(1-\overline{\varphi(z_k)}z\right)^{1+t+\frac{1}{q}}},k\in N,$$ 其中 $C_j=t+j+\frac{1}{q}$,$j=1,2$,$t$ 由正规函数 $\phi$ 确定. 那么 $h_k\in H(p,\,q,\,\phi)$,且 $\sup\limits_{k\in N}\|h_k\|_{p,\,q,\,\phi}\leq C.$又 $\{h_k\}$在 $D$的紧子集上一致收敛于0,且 $$h'_k(\varphi(z_k))=0,\,h''_k(\varphi(z_k))=\frac{C_{1}C_{2}\left(\overline{\varphi(z_k)}\right)^2}{\phi(|\varphi(z_k)|)\left(1-|\varphi(z_k)|^2\right)^{2+\frac{1}{q}}}.$$ 依引理2.2,算子 $I_{g,\varphi}$: $H(p,\,q,\,\phi)\rightarrow {\cal Z}_{0}$ 的紧性给出

$$\begin{equation} \label{e29}\lim_{k\rightarrow\infty}\|I_{g,\varphi}h_k\|_{\cal Z}=0.\end{equation}$$

(3.13)

另一方面,当 $k$ 充分大时, $$\begin{eqnarray*} \|I_{g,\varphi}h_k\|_{\cal Z} & \geq& \sup_{z\in D}(1-|z|^{2})|(I_{g,\varphi}h_k)''(z)|\\ &\geq& (1-|z_k|^{2})|\varphi'(z_k)|^2\cdot|g(\varphi(z_k))|\cdot|h_k''(\varphi(z_k))|\\ &&-(1-|z|^{2})|h'(\varphi(z_k))|\cdot|(\varphi'(z_k))^{2}g'(\varphi(z_k))+\varphi''(z_k)g(\varphi(z_k))|\\ &=&C_{1}C_{2}\frac{(1-|z_k|^{2})|\varphi'(z_k)|^2\cdot|\varphi(z_k)|^2\cdot|g(\varphi(z_k))|}{\phi(|\varphi(z_k)|)(1-|\varphi(z_k)|^{2})^{2+\frac{1}{q}}} \\ &\geq& \frac{C_{1}C_{2}}{2}\varepsilon_{0}. \end{eqnarray*}$$ 可以看出上式与(3.13)式矛盾. 同理,取检验函数 $$f_k(z)=\frac{(1-|\varphi(z_k)|^{2})^{t+1}}{\phi(|\varphi(z_k)|)\left(1-\overline{\varphi(z_k)}z\right)^{1+t+\frac{1}{q}}},\ k\in N,$$ 可证(3.5)式成立

$\bf 注1$ 由 (a) 推出 (3.4),(3.5) 式 成立也可直接证明. 这是因为 $I_{g,\varphi}$: $H(p,\,q,\,\phi)\rightarrow {\cal Z}_{0}$ 是紧算子, 则 $I_{g,\varphi}$: $H(p,\,q,\,\phi)\rightarrow {\cal Z}$ 是紧算子,根据文献[1,定理2]可知 (3.4),(3.5)式 成立.

$\bf 注2$ 由定理3.1及定理3.2可以看出,当 $\|\varphi\|_\infty =1$ 时,算子$I_{g,\varphi}$: $H(p,\,q,\,\phi)\rightarrow {\cal Z}_0$ 的有界性和 算子 $I_{g,\varphi}$: $H(p,\,q,\,\phi)\rightarrow {\cal Z}_0$ 的紧性并不等价.