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1、汕头大学 硕士学位论文 一类解析函数空间上的Toeplitz算子 姓名：姚炳洪 申请学位级别：硕士 专业：基础数学 指导教师：娄增建 2010-06 uE, 0, b L1, ToeplitzfXe: T (b)(z) = Z D (1 |w|2)1b(w) (1 wz)+1 d(w). Z. J. Wu, R. H. ZhaoN. Zorboska?ToeplitzfT 3Bloch.m k.;?dx. ?3a)maq?K, ) ?Toeplitzf3?k.5;5, (J2?Z. J. Wu, R. H. Zhao N. Zorboska?(“ ?. 1X-0?Toeplitzf, Bloc
2、h.ma )m?9?:k(. 1?|S?5?Toeplitzf3a)mk .;?dx. c: Toeplitzf, )m, Bloch.m. Abstract For a complex measure , 0, and b L1 , defi ne a Toeplitz operator as follows: T (b)(z) = Z D (1 |w|2)1b(w) (1 wz)+1 d(w). Z. J. Wu, R. H. Zhao and N. Zorboska characterized complex measurses for which the Toeplitz operat
3、or T is bounded or compact on Bloch-type spaces. In this thesis, we extend the result to a class of analytic functions and characterize complex measurses for which the Toeplitz operator T is bounded or compact on the space. The thesis is divided two parts. The fi rst part provides the defi nitions o
4、f Toeplitz operator, Bloch-type spaces and a class of analytic functions. The preliminaries and related known results are also provided. The second part gives necessary and suffi cient conditions for the Toeplitz operator T to be bounded or compact on the class of analytic functions by using propert
5、y of the inner product. Key Words.Toeplitz operator, analytic function spaces, Bloch-type spaces. ?M5( ? 0, b L1, ToeplitzfXe T (b)(z) = Z D (1 |w|2)1b(w) (1 wz)+1 d(w). ?u 1, E?BergmanKXe P()(z) = ( + 1) Z D (1 |w|2) (1 wz)+2 d(w). b L1?BergmanKXe P(b)(z) = ( + 1) Z D (1 |w|2)b(w) (1 wz)+2 dA(w). 3
6、 a 1.2. 12,14 u 0, ef H(D)v M(f) = sup zD |f0(z)|(1 |z|2) 0, o(H1,p)= H ,p0. f H1,p, g H ,p0, (f,g) = Z D f(z)g(z)(1 |z|2)1dA(z). AO/, (1) ef H1,p, g H ,p0, Tg(f) = (f,g), oTg (H1,p) kkTgk Ckgk,p0,. (2) ?, u?T (H1,p), 3?g H ,p0, ? T(f) = (f,g), f H1,pkgk,p0, CkTk. enToeplitzf3)mH ,pk.?dx. n2.2. e1 0
7、Ev R()(w) = (1 |w|2) Z D (1 |z|2)1 (z w)(1 wz)+1 d(z) L.(2.1) ok: (1) e0 0, z D, f H ,p. o f(z) = Z D ? f(0) + (1 |w|2)f0(w) w ? 1 (1 zw)+1 dA(w). y. ef H ,p, Z D (1 |w|2)|f0(w)|dA(w) 0? ? ?f(rei)? C(1 r)kfkBMOA. n2.4. eT (H1,p), 0 0, o Z 1 0 (1 r)1d C(1 r). y. -9n3.1? = 0, m = 1 + , =?(. n2.6. e1 p
8、 1.(2.13) e + 1 p = 1, K 1, odP1() H ,p?0 1, (2.7), (2.13)(2.15)An?, Mp(r,P 0 1()(1 r 2)2 L, 19 a -E?L?nv n 1 ? Mp(r,P 0 1()(1 r 2)n L. l?P1() Hn ,p, XP1() H 11 p ,p. y? n2.3?y: eyaqu12?n3.1. 75. u + 1 p = 1 + 1 p 1, -gnH ,p?S?v kgnk,p, 13D?;f8gn(w)u0. ?hH1,p0 ?. -w = rei, aqun2.2?y, k (h,T (gn) = Z
9、 D P(hgn)(z)(1 |z|2)1d(z) + Z D (I P)(hgn)(z)(1 |z|2)1d(z) = I1+ I2. u0 1. uw DDs Js?Cu1, ?+ 1 p = 1+ 1 p 1 Ok Mp(r,Q()(1 r2)log 2 1 r2 ?LP“5?%Y 3?K!?g?83?9 v?P“G%?h?AO2? !?!? j?) 3dL/a?“ g,a?=Md? ?r? =U?9ow?! X!U?!o?3?S?“ a?4?t! ?(9?9“7“*“3“3S) ?“ ?,ca?I1933a?3)S? %|?du?|U?%?/S|? “ %a?3zam?;?9F”? ? P“;“ 2007?): 2010c6? a ? 6 5OI)c?1985c8? ?m?x704X ;:mfn a S mS 2003.92007.7“? 2007.98 29
