周 林,曹成堂

(連云港廣播電視大學(xué) 建筑工程系,江蘇 連云港 222006)

0 引言

本文將要討論的算子如下:

1 預(yù)備引理

引理1[5]設(shè)f∈B,那么對于任意正整數(shù)n,

‖f‖B?∑|f(j)(0)|+

引理2[5]在B0中的一個(gè)閉子集K是緊的充要條件是K是有界的且滿足

由Montel定理及緊算子定義,可以得出下面的引理.

2 主要定理及證明

(1)

證明首先假設(shè)(1)成立,那么對于任意f∈B,由引理1可得

(1-|z|2)|u(z)f(n)(φ(z))|≤

C‖

(2)

所以

C<∞.

(3)

(4)

又因?yàn)?/p>

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C<∞.

由上式以及(3),(4)可得(1)成立.

(5)

(6)

(7)

由(6)以及(7)可知,當(dāng)i>i0時(shí),有

下證‖φ‖∞=1的情況.令(zi)i∈∈D是使得|φ(zi)|→1,i→∞的點(diǎn)列.

(8)

(1-|z|2)|u(z)p(n)(φ(z))|≤

(1-|z|2)|u(z)|‖p(n)(φ(z))‖B.

(9)

證明首先假設(shè)(9)成立.在(2)中對‖f‖B≤1的f取上確界,且讓|z|→1,可得

(10)

由(8)成立可得,?δ∈(0,1),當(dāng)δ<|z|<1時(shí),

(1-|z|2)|u(z)|≤ε(1-r2)n.

(11)

由(10)可得,當(dāng)δ<|z|<1,r<|φ(z)|<1時(shí),

(12)

由(11)可得,當(dāng)δ<|z|<1,|φ(z)|≤r時(shí),

(13)

由(12)以及(13)可得(9)成立.

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