1995年Roper-Suffridge算子的引入, 为在多复变中构造具有特殊几何性质的双全纯映照提供了强有力工具, 因此许多人都致力于研究它.近年来在不同空间及不同域上对Roper-Suffridge算子的推广及其性质的讨论有许多很好的结果^[1-8].

2005年刘名生和朱玉灿将Roper-Suffridge算子推广为

$ F(x)=\Phi_{\beta_{2}, \cdot\cdot\cdot, \beta_{n-1}, 0}(f)(x)=\sum\limits_{j=1}^{n-1}(\displaystyle\frac{f(T_{x_{1}}(x))}{T_{x_{1}}(x)})^{\beta_{j}} T_{x_{j}}(x)x_{j}+x-\sum\limits_{j=1}^{n-1}T_{x_{j}}(x)x_{j}, $

并讨论了该算子的一些性质^[9].

2010年邹娟^[10]定义了算子

$ \phi_{\beta, \gamma}(f)(x)=f(T_{x_{1}}(x))x_{1}+(\displaystyle\frac{f(T_{x_{1}}(x))}{T_{x_{1}}(x)})^{\beta} (f^{'}(T_{x_{1}}(x)))^{\gamma}(x-T_{x_{1}}(x)x_{1}). $

并证明了该算子在一定区域上保持$\beta$型螺型性质以及当$\gamma=0$时保持$\alpha$次星形性.

本文旨在讨论文献[5]中的算子保持双全纯映照子族的其它性质.在全文中, $D$表示单位圆盘, $X$为复Banach空间, $B=\{x\in X:\|x\|<1\}$表示$X$中单位球, $X^{\ast}$是$X$的对偶空间, 对任意的$x\in X\backslash\{0\}$, $T_{x}=\{T_{x}\in X^{\ast}:\|T_{x}\|=1, T_{x}(x)=\|x\|\}$为连续线性泛函, 由Hahn-Banach定理知此集合非空.

定义 1.1^[11] 若$f$同上, $\rho\in{[0, 1)}$, $\beta\in{(-\displaystyle\frac{\pi}{2}, \displaystyle\frac{\pi}{2})}$,

$ |\displaystyle\frac{e^{-i\beta}}{\|x\|}T_{x}[(Df(x))^{-1}f(x)]-(1-i\sin\beta)| <(1-2\rho)+{\hbox{Re}}\{\displaystyle\frac{e^{-i\beta}}{\|x\|}T_{x}[(Df(x))^{-1}f(x)]\}, \quad\\ x\in B, $

则称$f$是$B$上$\rho$次抛物形$\beta$型螺形映照.

1995年Chuaqui建立了$C^{n}$中单位球$B^{n}$上强$\alpha$次殆星形映照^[12]的概念, 后来刘小松将其推广到复Banach空间单位球$B$上:

定义 1.2 ^[13] 若$f$同上, $\alpha\in{[0, 1)}$, $c\in{(0, 1)}$,

$ |\displaystyle\frac{1}{1-\alpha}\{\displaystyle\frac{1}{\|x\|}T_{x}[(Df(x))^{-1}f(x)]-\alpha\}-\displaystyle\frac {1+c^{2}}{1-c^{2}}|<\displaystyle\frac{2c}{1-c^{2}}, \quad x\in B\backslash\{0\}, $

则称$f$是$B$上强$\alpha$次殆星形映照.

引理 2.1 ^[9] 如果$f$是$D$上正规化双全纯映照, 则

$ F(x)=\Phi_{\beta_{2}, \cdots, \beta_{n-1}, 0}(f)(x)=\sum\limits_{j=1}^{n-1}(\displaystyle\frac{f(T_{x_{1}}(x))}{T_{x_{1}}(x)})^{\beta_{j}} T_{x_{j}}(x)x_{j}+x-\sum\limits_{j=1}^{n-1}T_{x_{j}}(x)x_{j} $

是$B$上正规化双全纯映照, 其中$n\in N(n\geq2), \beta_{1}=1, 0\leq\beta_{j}\leq1, \quad j=2, \cdots, n-1$, 且$(\displaystyle\frac{f(z)}{z})^{\beta_{j}}|_{z=0}=1$, $j=1, \cdots, n-1$, $x_{1}\in\bar B$, $\|x_{1}\|=1$, 并且$x_{1}, \cdots, x_{n}\in X$线性无关, 对任意$x_{i}$, 选取$T_{x_{i}}\in X^{\ast}$, 使$\|T_{x_{i}}\|=1$, 且$T_{x_{i}}(x_{i})=1$, $T_{x_{i}}(x_{j})=0~~(i\neq j)$(由Hahn-Banach定理及其推论知此条件可取到).

引理 2.2 ^[10] 如果$x_{1}\in\bar B$, $\|x_{1}\|=1$, 则存在$T_{x_{1}}\in T(x_{1})$, $T_{x}\in T(x)$, 使得$\|x\|T_{x}(x_{1})=\overline{T_{x_{1}}(x)}. $

引理 2.3  $f$是$D$上正规化双全纯映照, $F(x)$为引理2.1中定义的函数, 则

$ \|x\|T_{x}[(DF(x))^{-1}F(x)]=\|x\|^{2}+\sum\limits_{j=1}^{n-1}\beta_{j}|T_{x_{j}}(x)|^{2} [\displaystyle\frac{f(T_{x_{1}}(x))}{T_{x_{1}}(x)f^{'}(T_{x_{1}}(x))}-1]. $

证  令$\omega=F(x)=\Phi_{\beta_{2}, \cdots, \beta_{n-1}, 0}(f)(x), $则$\omega$关于$x$全纯, 且

$ T_{x_{1}}(\omega)=(\displaystyle\frac{f(T_{x_{1}}(x))}{T_{x_{1}}(x)})^{1}T_{x_{1}}(x)T_{x_{1}}(x_{1}) +T_{x_{1}}(x)-T_{x_{1}}(x)=f(T_{x_{1}}(x)). $

(1)

同理有

$ T_{x_{j}}(\omega)=(\displaystyle\frac{f(T_{x_{1}}(x))}{T_{x_{1}}(x)})^{\beta_{j}}T_{x_{j}}(x), $

(2)

又$f$是$D$上正规化双全纯映照, 则由式(1) 及(2) 知

$ T_{x_{1}}(x)=f^{-1}(T_{x_{1}}(\omega)), $

(3)

$ T_{x_{j}}(x)=(\displaystyle\frac{T_{x_{1}}(\omega)}{f^{-1}(T_{x_{1}}(\omega))})^{-\beta_{j}}T_{x_{j}}(\omega), $

(4)

将式(3) 及(4) 代入$\omega=F(x)=\Phi_{\beta_{2}, \cdots, \beta_{n-1}, 0}(f)(x)$得

$ \omega=\sum\limits_{j=1}^{n-1}T_{x_{j}}(\omega)x_{j}+ [x-\sum\limits_{j=1}^{n-1}(\displaystyle\frac{T_{x_{1}}(\omega)}{f^{-1}(T_{x_{1}}(\omega))})^{-\beta_{j}}T_{x_{j}}(\omega)x_{j}], $

$ x=(\omega-\sum\limits_{j=1}^{n-1}T_{x_{j}}(\omega)x_{j})+ \sum\limits_{j=1}^{n-1}(\displaystyle\frac{T_{x_{1}}(\omega)}{f^{-1}(T_{x_{1}}(\omega))})^{-\beta_{j}}T_{x_{j}}(\omega)x_{j}, $

(5)

由式(5) 知

$ \begin{eqnarray*}&& x=F^{-1}(\omega)=(\omega-\sum\limits_{j=1}^{n-1}T_{x_{j}}(\omega)x_{j})+ \sum\limits_{j=1}^{n-1}(\displaystyle\frac{T_{x_{1}}(\omega)}{f^{-1}(T_{x_{1}}(\omega))})^{-\beta_{j}}T_{x_{j}}(\omega)x_{j}, \\ && (DF(x))^{-1}\eta=DF^{-1}(\omega)\eta=(\eta-\displaystyle\sum\limits_{j=1}^{n-1}T_{x_{j}}(\eta)x_{j})\\ && +\displaystyle\sum\limits_{j=1}^{n-1}\{(-\beta_{j})(\displaystyle\frac{T_{x_{1}}(\omega)}{f^{-1}(T_{x_{1}}(\omega))})^{-\beta_{j}-1} \displaystyle\frac{T_{x_{1}}(\eta)f^{-1}(T_{x_{1}}(\omega))-T_{x_{1}}(\omega)(f^{-1})^{'}(T_{x_{1}}(\omega)) T_{x_{1}}(\eta)}{[f^{-1}(T_{x_{1}}(\omega))]^{2}}\\ && \cdot T_{x_{j}}(\omega)x_{j}+(\displaystyle\frac{T_{x_{1}}(\omega)}{f^{-1}(T_{x_{1}}(\omega))})^{-\beta_{j}}T_{x_{j}}(\eta)x_{j})\}. \end{eqnarray*} $

由式(4) 及(5) 知

$ \omega-\sum\limits_{j=1}^{n-1}T_{x_{j}}(\omega)x_{j}=x-\sum\limits_{j=1}^{n-1}(\displaystyle\frac{T_{x_{1}}(\omega)}{f^{-1}(T_{x_{1}}(\omega))})^{-\beta_{j}}T_{x_{j}}(\omega)x_{j} =x-\sum\limits_{j=1}^{n-1}T_{x_{j}}(x)x_{j}, $

(6)

由式(3) 知

$ (f^{-1})^{'}(T_{x_{1}}(\omega))=\displaystyle\frac{1}{f^{'}(T_{x_{1}}(x))}, $

(7)

则由式(1)、(3)、(4)、(6)、(7) 知

$ \begin{eqnarray*}&& (DF(x))^{-1}F(x)=x-\displaystyle\sum\limits_{j=1}^{n-1}T_{x_{j}}(x)x_{j}\\ && +\displaystyle\sum\limits_{j=1}^{n-1}\{(-\beta_{j})(\displaystyle\frac{T_{x_{1}}(\omega)}{f^{-1}(T_{x_{1}}(\omega))})^{-\beta_{j}} \displaystyle\frac{f^{-1}(T_{x_{1}}(\omega))-(f^{-1})^{'}(T_{x_{1}}(\omega)) T_{x_{1}}(\omega)}{f^{-1}(T_{x_{1}}(\omega))}T_{x_{j}}(\omega)x_{j}\\ &&+(\displaystyle\frac{T_{x_{1}}(\omega)}{f^{-1}(T_{x_{1}}(\omega))})^{-\beta_{j}}T_{x_{j}}(\omega)x_{j}\}\\ &=&\; x-\displaystyle\sum\limits_{j=1}^{n-1}T_{x_{j}}(x)x_{j}+\displaystyle\sum\limits_{j=1}^{n-1}\{(\displaystyle\frac{T_{x_{1}}(\omega)}{f^{-1}(T_{x_{1}}(\omega))})^{-\beta_{j}}T_{x_{j}}(\omega)x_{j} \{1+\beta_{j}[\displaystyle\frac{f(T_{x_{1}}(x))}{T_{x_{1}}(x)f^{'}(T_{x_{1}}(x))}-1]\}\}\\ &=&\; x-\displaystyle\sum\limits_{j=1}^{n-1}T_{x_{j}}(x)x_{j}+\displaystyle\sum\limits_{j=1}^{n-1}\{T_{x_{j}}(x)x_{j} \{1+\beta_{j}[\displaystyle\frac{f(T_{x_{1}}(x))}{T_{x_{1}}(x)f^{'}(T_{x_{1}}(x))}-1]\}\}\\ &=&\; x+\displaystyle\sum\limits_{j=1}^{n-1}\beta_{j}T_{x_{j}}(x)x_{j} [\displaystyle\frac{f(T_{x_{1}}(x))}{T_{x_{1}}(x)f^{'}(T_{x_{1}}(x))}-1], \end{eqnarray*} $

由引理2.2知

$ \begin{eqnarray*}&& \|x\|T_{x}[(DF(x))^{-1}F(x)]\\ & =&\; \|x\|^{2}+\displaystyle\sum\limits_{j=1}^{n-1}\beta_{j}T_{x_{j}}(x)\|x\|T_{x}(x_{j}) [\displaystyle\frac{f(T_{x_{1}}(x))}{T_{x_{1}}(x)f^{'}(T_{x_{1}}(x))}-1]\\ &=&\;\|x\|^{2}+\displaystyle\sum\limits_{j=1}^{n-1}\beta_{j}|T_{x_{j}}(x)|^{2}[\displaystyle\frac{f(T_{x_{1}}(x))}{T_{x_{1}}(x)f^{'}(T_{x_{1}}(x))}-1]. \end{eqnarray*} $

定理 2.4  $f$是$D$上$\rho$次抛物形$\beta$型螺形映照, $\rho\in{[0, 1)}$, $\cos\beta>\rho$, $F(x)$为引理2.1中定义的函数, 则$F(x)$是$B$上$\rho$次抛物形$\beta$型螺形映照.

证  由于$f$是$D$上$\rho$次抛物形$\beta$型螺形映照, 则

$ |e^{-i\beta}\displaystyle\frac{f(T_{x_{1}}(x))}{T_{x_{1}}(x)f^{'}(T_{x_{1}}(x))}-(1-i\sin\beta)| <(1-2\rho)+{\hbox {Re}}\{e^{-i\beta}\displaystyle\frac{f(T_{x_{1}}(x))}{T_{x_{1}}(x)f^{'}(T_{x_{1}}(x))}\}, $

则由引理2.3知

$ \;\;\;\;|\displaystyle\frac{e^{-i\beta}}{\|x\|}T_{x}[(Df(x))^{-1}f(x)]-(1-i\sin\beta)|\\ =\; |e^{-i\beta}\{1+\displaystyle\frac{\displaystyle\sum\limits_{j=1}^{n-1}\beta_{j}|T_{x_{j}}(x)|^{2}}{\|x\|^{2}} [\displaystyle\frac{f(T_{x_{1}}(x))}{T_{x_{1}}(x)f^{'}(T_{x_{1}}(x))}-1]\}-(1-i\sin\beta)|\\ =\; |e^{-i\beta}(1-\displaystyle\frac{\displaystyle\sum\limits_{j=1}^{n-1}\beta_{j}|T_{x_{j}}(x)|^{2}}{\|x\|^{2}}) +\displaystyle\frac{\displaystyle\sum\limits_{j=1}^{n-1}\beta_{j}|T_{x_{j}}(x)|^{2}}{\|x\|^{2}} [e^{-i\beta}\displaystyle\frac{f(T_{x_{1}}(x))}{T_{x_{1}}(x)f^{'}(T_{x_{1}}(x))}-(1-i\sin\beta)]\\ \;\;\;\;+(\displaystyle\frac{\displaystyle\sum\limits_{j=1}^{n-1}\beta_{j}|T_{x_{j}}(x)|^{2}}{\|x\|^{2}}-1)(1-i\sin\beta)|\\ =\; |(1-\displaystyle\frac{\displaystyle\sum\limits_{j=1}^{n-1}\beta_{j}|T_{x_{j}}(x)|^{2}}{\|x\|^{2}})(\cos\beta-1) +\displaystyle\frac{\displaystyle\sum\limits_{j=1}^{n-1}\beta_{j}|T_{x_{j}}(x)|^{2}}{\|x\|^{2}} [e^{-i\beta}\displaystyle\frac{f(T_{x_{1}}(x))}{T_{x_{1}}(x)f^{'}(T_{x_{1}}(x))}\\ -(1-i\sin\beta)]|\\ \leq\;(1-\displaystyle\frac{\displaystyle\sum\limits_{j=1}^{n-1}\beta_{j}|T_{x_{j}}(x)|^{2}}{\|x\|^{2}})(1-\cos\beta) +\displaystyle\frac{\displaystyle\sum\limits_{j=1}^{n-1}\beta_{j}|T_{x_{j}}(x)|^{2}}{\|x\|^{2}} |e^{-i\beta}\displaystyle\frac{f(T_{x_{1}}(x))}{T_{x_{1}}(x)f^{'}(T_{x_{1}}(x))}\\ -(1-i\sin\beta)|\\ <\;(1-\displaystyle\frac{\displaystyle\sum\limits_{j=1}^{n-1}\beta_{j}|T_{x_{j}}(x)|^{2}}{\|x\|^{2}})(1-\cos\beta) +\displaystyle\frac{\displaystyle\sum\limits_{j=1}^{n-1}\beta_{j}|T_{x_{j}}(x)|^{2}}{\|x\|^{2}} [(1-2\rho)+\\Re\{e^{-i\beta}\displaystyle\frac{f(T_{x_{1}}(x))}{T_{x_{1}}(x)f^{'}(T_{x_{1}}(x))}\}]\\ =\;(1-\displaystyle\frac{\displaystyle\sum\limits_{j=1}^{n-1}\beta_{j}|T_{x_{j}}(x)|^{2}}{\|x\|^{2}})(1-2\cos\beta) +\displaystyle\frac{\displaystyle\sum\limits_{j=1}^{n-1}\beta_{j}|T_{x_{j}}(x)|^{2}}{\|x\|^{2}} (1-2\rho)\\ \;\;\;\;+{\hbox{Re}}\{e^{-i\beta}\{1+\displaystyle\frac{\displaystyle\sum\limits_{j=1}^{n-1}\beta_{j}|T_{x_{j}}(x)|^{2}}{\|x\|^{2}} [\displaystyle\frac{f(T_{x_{1}}(x))}{T_{x_{1}}(x)f^{'}(T_{x_{1}}(x))}-1]\}\}\\ <\;(1-\displaystyle\frac{\displaystyle\sum\limits_{j=1}^{n-1}\beta_{j}|T_{x_{j}}(x)|^{2}}{\|x\|^{2}})(1-2\rho) +\displaystyle\frac{\displaystyle\sum\limits_{j=1}^{n-1}\beta_{j}|T_{x_{j}}(x)|^{2}}{\|x\|^{2}} (1-2\rho)\\ \;\;\;\;+{\hbox{Re}}\{e^{-i\beta}\{1+\displaystyle\frac{\displaystyle\sum\limits_{j=1}^{n-1}\beta_{j}|T_{x_{j}}(x)|^{2}}{\|x\|^{2}} [\displaystyle\frac{f(T_{x_{1}}(x))}{T_{x_{1}}(x)f^{'}(T_{x_{1}}(x))}-1]\}\}\\ =\;(1-2\rho)+{\hbox{Re}}\{\displaystyle\frac{e^{-i\beta}}{\|x\|}T_{x}[(Df(x))^{-1}f(x)]\}. $

由定义1.1知$F(x)$是$B$上$\rho$次抛物形$\beta$型螺形映照.

推论 2.5  $f$是$D$上$\rho$次抛物星形映照, $\rho\in{[0, 1)}$, $\cos\beta>\rho$, $F(x)$为引理2.1中定义的函数, 则$F(x)$是$B$上$\rho$次抛物星形映照.

定理 2.6  $f$是$D$上强$\alpha$次殆星形映照, $\alpha\in{[0, 1)}$, $c\in{(0, 1)}$, $F(x)$为引理2.1中定义的函数, 则$F(x)$是$B$上强$\alpha$次殆星形映照.

证

$ \begin{eqnarray*}&& |\displaystyle\frac{1}{1-\alpha}\{\displaystyle\frac{1}{\|x\|}T_{x}[(Df(x))^{-1}f(x)]-\alpha\}-\displaystyle\frac {1+c^{2}}{1-c^{2}}|\\ &\;=&\;|\displaystyle\frac{1}{1-\alpha}\{1+\displaystyle\frac{\displaystyle\sum\limits_{j=1}^{n-1}\beta_{j}|T_{x_{j}}(x)|^{2}} {\|x\|^{2}}[\displaystyle\frac{f(T_{x_{1}}(x))}{T_{x_{1}}(x)f^{'}(T_{x_{1}}(x))}-1]-\alpha\}-\displaystyle\frac {1+c^{2}}{1-c^{2}}|\\ &=&\;\;\;\;|\displaystyle\frac{1}{1-\alpha}\displaystyle\frac{\displaystyle\sum\limits_{j=1}^{n-1}\beta_{j}|T_{x_{j}}(x)|^{2}} {\|x\|^{2}}[\displaystyle\frac{f(T_{x_{1}}(x))}{T_{x_{1}}(x)f^{'}(T_{x_{1}}(x))}-1]+1-\displaystyle\frac {1+c^{2}}{1-c^{2}}|\\ &=&\;\;\;\;|\displaystyle\frac{\displaystyle\sum\limits_{j=1}^{n-1}\beta_{j}|T_{x_{j}}(x)|^{2}} {\|x\|^{2}}\{\displaystyle\frac{1}{1-\alpha}[\displaystyle\frac{f(T_{x_{1}}(x))}{T_{x_{1}}(x)f^{'}(T_{x_{1}}(x))}-\alpha] -\displaystyle\frac{1+c^{2}}{1-c^{2}}\}\\ &&+(1-\displaystyle\frac{\displaystyle\sum\limits_{j=1}^{n-1}\beta_{j}|T_{x_{j}}(x)|^{2}} {\|x\|^{2}})(1-\displaystyle\frac{1+c^{2}}{1-c^{2}})|\\ &<&\;\;\;\;\displaystyle\frac{\displaystyle\sum\limits_{j=1}^{n-1}\beta_{j}|T_{x_{j}}(x)|^{2}} {\|x\|^{2}}\displaystyle\frac{2c}{1-c^{2}}+(1-\displaystyle\frac{\displaystyle\sum\limits_{j=1}^{n-1}\beta_{j}|T_{x_{j}}(x)|^{2}} {\|x\|^{2}})\displaystyle\frac{2c}{1-c^{2}} =\displaystyle\frac{2c}{1-c^{2}}.\end{eqnarray*} $

由定义1.2知$F(x)$是$B$上强$\alpha$次殆星形映照, 当$\alpha=0$时即为强星形映照.

注  若$X$是$n$维复Hilbert空间, 则由Riesz表示定理知$T_{x_{1}}(x)=\langle x, x_{1}\rangle$, 取$x_{1}=(1, \cdot\cdot\cdot, 0)$, 则$\|x\|=1$, $x=(z_{1}, \cdot\cdot\cdot, z_{n})=(z_{1}, z_{0})$, 则有$T_{x_{1}}(x)=z_{1}$, 于是引理2.1所定义的函数为:

$ F(z)=(f(z_{1}), (\displaystyle\frac{f(z_{1})}{z_{1}})^{\beta_{2}}z_{2}, \cdot\cdot\cdot, (\displaystyle\frac{f(z_{1})}{z_{1}})^{\beta_{n-1}}z_{n-1}, z_{n}). $

(8)

特别当$n=2$时有$F(z)=(f(z_{1}), z_{2}).$

推论 2.7  令$F(z)$为式(8) 定义的函数, 其中$n\in N(n\geq2), \beta_{1}=1, 0\leq\beta_{j}\leq1, \quad j=2, \cdot\cdot\cdot, n-1$, 且$(\displaystyle\frac{f(z)}{z})^{\beta_{j}}|_{z=0}=1$ $(j=1, \cdot\cdot\cdot, n-1)$, 则$F(x)$在复Hilbert空间单位球$B^{n}$上保持$\rho$次抛物形$\beta$型螺形映照及强$\alpha$次殆星形映照.