在这一节中,我们对从$\ F$(p,q,s) (或${{\ F}_0}$(p,q,s))到${\cal B}^{\mu}$ (或$B_0^\mu)$上的 广义复合算子的有界性、 紧性进行刻画,首先介绍下面引理.

引理3.1 ^[17] 设$0<p,s<\infty$,$-2<q<\infty$,$q+s>-1$并且$f\in\ F$(p,q,s), 则存在常数$C>0$使得$\| f\| _{{\beta ^\alpha }}\leq C\| f\| _{F}$,此外,如果$f\in{{\ F}_0}$(p,q,s), 则$f\in\beta _0^\alpha$. 这里$\alpha=\frac{q+2}{p}$.

类似引理2.1,我们易证下面的结论.

引理3.2 设$g\in H({\Bbb D})$,$\varphi:{\Bbb D}\rightarrow{\Bbb D}$解析,$\mu$是正规权. 又设$0<p,s<\infty$,$-2<q<\infty$,$q+s>-1$. 则$C_{\varphi}^{g}:\ F$ (p,q,s)(或$\;{F_0}(p,q,s) \to {B^\mu }$)(或$B_0^\mu)$ 是紧算子当且仅当$C_\varphi ^g:F(p,q,s)$(或${F_0}(p,q,s) \to {B^\mu }$ (或$B_0^\mu)$ 有界并且对于${\ F}(p,q,s)$ (或${{\ F}_0}(p,q,s))$中的任意有界序列$\{f_{n}\}_{n\in\mathbb{N}}$, $f_{n}$在${\Bbb D}$ 的紧子集上一致收敛于$0$时,有$\| C_{\varphi}^{g}f_{n}\| _{{B^\mu }}\rightarrow0$.

定理3.1 设$g\in H({\Bbb D})$,$\varphi:{\Bbb D}\rightarrow{\Bbb D}$解析,$\mu$是正规权. 又设$0<p,s<\infty$, $-2<q<\infty$,$q+s>-1$. 则$C_\varphi ^g:F{\text{(p,q,s)}} \to {B^\mu }$有界当且仅当

证 设$C_\varphi ^g:F(p,q,s) \to {B^\mu }$有界. 令$\alpha=\frac{q+2}{p}$,固定$w\in{\Bbb D}$, 考虑函数

反之,设(3.1)式成立. 对任意$f \in F(p,q,s)$,根据引理3.1有 $$\begin{eqnarray*} \mathop {\sup }\limits_{z \in \mathbb{D}} {(1 - |z{|^2})^\alpha }|f'(z)| \leqslant C{\left\| f \right\|_{F{\text{(p,q,s)}}}} \end{eqnarray*}$$ 则 $$\begin{eqnarray*} {\left\| {C_\varphi ^gf(z)} \right\|_{{B^\mu }}} = \mathop {\sup }\limits_{z \in \mathbb{D}} \mu (|z|)|f'(\varphi (z))\left\| {g(z)|} \right. \hfill \\ = \mathop {\sup }\limits_{z \in \mathbb{D}} {(1 - |\varphi (z){|^2})^\alpha }|f'(\varphi (z))|\frac{{\mu (|z|)|g(z)|}}{{{{(1 - |\varphi (z){|^2})}^\alpha }}} \hfill \\ \leqslant C{\left\| f \right\|_{F{\text{(p,q,s)}}}}\mathop {\sup }\limits_{z \in \mathbb{D}} \frac{{\mu (|z|)|g(z)|}}{{{{(1 - |\varphi (z){|^2})}^\alpha }}} \hfill \\ \end{eqnarray*}$$ 由(3.1)式知$C_\varphi ^g:F{\text{(p,q,s)}} \to {B^\mu }$有界.

定理3.2 设$g\in H({\Bbb D})$,$\varphi:{\Bbb D}\rightarrow{\Bbb D}$解析,$\mu$是正规权. 又设$0<p,s<\infty$, $-2<q<\infty$,$q+s>-1$. 则$C_\varphi ^g:F{\text{(p,q,s)}} \to {B^\mu }$是紧算子当且仅当

证 设$C_\varphi ^g:F{\text{(p,q,s)}} \to {B^\mu }$是紧算子. 令$\alpha=\frac{q+2}{p}$, 并且$\{z_n\}_{z\in\mathbb{N}}$是${\Bbb D}$中的序列满足当$n\rightarrow\infty$时,$|\varphi(z_{n})|\rightarrow1$. 考虑函数 $$f_{n}(z)=\frac{(z-\varphi(z_{n}))(1-|\varphi(z_{n})|^{2})}{(1-\overline{\varphi(z_{n})}z)^{\alpha+1}}.$$ 同样用文献^[16]中的方法得到${f_n} \in F{\text{(p,q,s)}}$,$(f_{n})_{n\in\mathbb{N}}$是$F(p,q,s)$中 的一个有界序列,并且在${\Bbb D}$的紧子集上一致收敛于$0$. 注意到$f_{n}(\varphi(z_n))=0$,$f'(\varphi(z_n))=\frac{1}{(1-|\varphi(z_n)|^2)^{\alpha}}$. 即有 $$\frac{\mu(|z_{n}|)|g(z_{n})|}{(1-|\varphi(z_{n})|^{2})^{\alpha}}=\mu(|z_{n}|)|f'(\varphi(z_{n}))\| g(z_{n})|\leq\| C_{\varphi}^{g}f_{n}\| _{{B^\mu }}.$$ 根据引理3.2,(3.3)式成立.

反过来,设(3.3)式成立. 令$\{f_{n}\}_{n\in\mathbb{N}}$是$\ F(p,q,s)$中有界序列并在${\Bbb D}$的紧 子集上一致收敛于$0$. 令$M = \mathop {\sup }\limits_n {\left\| {{f_n}} \right\|_{F{\text{(p,q,s)}}}}<\infty$,根据引理3.2, 往证$\lim\limits_{n\rightarrow\infty}\| C_{\varphi}^{g}f_{n}\| _{{B^\mu }}=0$. 由(3.3)式,对于任意$\varepsilon>0$,存在$r\in(0,1)$,当$|\varphi(z)|>r$时, 有$\frac{\mu(|z|)|g(z)|}{(1-|\varphi(z)|^{2})^{\alpha}}<\varepsilon$. 由Cauchy 估计, 如果$\{f_{n}\}_{n\in\mathbb{N}}$在${\Bbb D}$的紧子集上一致收敛于$0$,则$\{f'_{n}\}_{n\in\mathbb{N}}$ 在${\Bbb D}$的紧子集上也一致收敛于$0$,即存在$n_{0}>0$使得当$n>n_{0}$ 时, $\sup\limits_{\{|\varphi(w)|\leq r\}}|f'_{n}(\varphi(w))|<\varepsilon$. 由以上叙述及引理3.1得到 $$\begin{eqnarray*} & &\sup\limits_{w\in{\Bbb D}}\mu(|w|)|f'_{n}(\varphi(w))\| g(w)| \\ & \leq&\sup\limits_{\{|\varphi(w)|\leq r\}}\mu(|w|)|f'_{n}(\varphi(w))\| g(w)| +\sup\limits_{\{|\varphi(w)|>r\}}\mu(|w|)|f'_{n}(\varphi(w))\| g(w)|\\ & =& \sup\limits_{\{|\varphi(w)|\leq r\}}\mu(|w|)|f'_{n}(\varphi(w))\| g(w)|\\ & &+ \sup\limits_{\{|\varphi(w)|>r\}}(1-|\varphi(w)|^2)^{\alpha}|f'_{n}(\varphi(w))|\frac{\mu(|w|)|g(w)|}{(1-|\varphi(w)|^2)^{\alpha}}\\ &\leq &C\sup\limits_{\{|\varphi(w)|\leq r\}}|f_{n}'(\varphi(w))|+\| f_{n}\| _{\ F}\sup\limits_{\{|\varphi(w)|>r\}}\frac{\mu(|w|)|g(w)|}{(1-|\varphi(w)|^{2})^{\alpha}}\\ & \leq& (C+M)\varepsilon, \end{eqnarray*}$$ 即当$n\rightarrow\infty$时,$\| C_{\varphi}^{g}f_{n}\| _{{B^\mu }}=\sup\limits_{w\in{\Bbb D}}\mu(|w|)|f'(\varphi(w))\| g(w)|\rightarrow0$,证明完成.

定理3.3 设$g\in H({\Bbb D})$,$\varphi:{\Bbb D}\rightarrow{\Bbb D}$解析,$\mu$是正规权. 又设$0<p,s<\infty$, $-2<q<\infty$,$q+s>-1$. 则$C_\varphi ^g:{F_{{\text{(p,q,s)}}}} \to {B^\mu }$有界当且仅当 $C_\varphi ^g:{F_{{\text{(p,q,s)}}}} \to {B^\mu }$有界并且

证 设$C_\varphi ^g:{F_{{\text{(p,q,s)}}}} \to {B^\mu }$有界,则$C_\varphi ^g:{F_{{\text{(p,q,s)}}}} \to {B^\mu }$有界. 取$f(z) = z \in {F_0}{\text{(p,q,s)}}$,即得到(3.4)式.

反之,设$C_\varphi ^g:{F_{{\text{(p,q,s)}}}} \to {B^\mu }$有界并且(3.4)式成立. 则对任意多项式$P$有

下面我们给出在满足一定条件下小空间与对应空间上的广义复合算子的紧性是等价的.

定理3.4 设$g\in H({\Bbb D})$,$\varphi:{\Bbb D}\rightarrow{\Bbb D}$解析,$\mu$是正规权. 又设$0<p,s<\infty$, $-2<q<\infty$,$q+s>-1$. 则下面条件等价.

(1) $C_\varphi ^g:F{\text{(p,q,s)}} \to {B^\mu }$是紧算子.

(2) $C_\varphi ^g:F{\text{(p,q,s)}} \to {B^\mu }$是紧算子.

(3)

证 $(1)\Rightarrow(2)$ 显然.

$(2)\Rightarrow(3)$ 令$\alpha=\frac{q+2}{p}$. 设$C_\varphi ^g:F{\text{(p,q,s)}} \to {B^\mu }$是紧算子, 由引理2.2,对$M>0$有 $$\lim\limits_{|z|\rightarrow1}\sup\limits_{\| f\| _{\ F\leq M,f\in{{\ F}_0}}} \mu(|z|)|(C_{\varphi}^{g}f)'(z)|=0.$$ 注意到在定理3.1的证明中给出的函数$f_{w}(z)=\frac{(z-\varphi(w))(1-|\varphi(w)|^{2})}{(1-\overline{\varphi(w)}z)^{\alpha+1}}$属于${{\ F}_0}$, 并且关于$w$范数一致有界,则 $$\sup\limits_{|w|\rightarrow1}\frac{\mu(|w|)|g(w)|}{(1-|\varphi(w)|^{2})^{\alpha}}=0.$$ 即得(3.6)式成立.

$(3)\Rightarrow(1)$ 设$f\in\ F$满足$\| f\| _{\ F}\leq1$. 由引理3.1, $f\in{\cal B}^{\alpha}$,$\| f\| _{{\cal B}^{\alpha}}\leq C\| f\| _{\ F}$. $$\begin{eqnarray*} \mu(|z|)|(C_{\varphi}^{g}f)'(z)|&=&\mu(|z|)|f'(\varphi(z))\| g(z)|\\ &=&(1-|\varphi(z)|^2)^{\alpha}|f'(\varphi(z))|\frac{\mu(|z|)|g(z)|}{(1-|\varphi(z)|^2)^{\alpha}} \\ & \leq &C\| f\| _{\ F}\frac{\mu(|z|)|g(z)|}{(1-|\varphi(z)|^{2})^{\alpha}}. \end{eqnarray*}$$ 由(3.6)式得 $\lim\limits_{|z|\rightarrow1}\sup\limits_{\| f\| _{\ F\leq1}} \mu(|z|)|(C_{\varphi}^{g}f)'(z)|=0.$ 由引理2.2知$C_\varphi ^g:F{\text{(p,q,s)}} \to {B^\mu }$ 是紧算子.