Let φ be a linear fractional self-map of the ball BN with a boundary fixed point e1,we show that1φReφ1(z)～Re(1-z1)holds in a neighborhood of e1 on BN.Applying this result we give a positive answer for a conjecture by MacCluer and Weir,and improve their results relating to the essential normality of composition operators on H 2(BN)and A2 γ (BN)(γ>-1).Combining this with other related results in MacCluer& Weir,Integral Equations Operator Theory,2005,we characterize the essential normality of composition operators induced by parabolic or hyperbolic linear fractional self-maps of B2.Some of them indicate a difference between one variable and several variables.
JIANG LiangYing1,2& OUYANG CaiHeng3 1Department of Mathematics,Tongji University,Shanghai 200092,China 2Department of Applied Mathematics,Shanghai Finance University,Shanghai 201209,China 3Wuhan Institute of Physics and Mathematics,Chinese Academy of Sciences,Wuhan 430071,China
Suppose f is a spirallike function of type β and order α on the unit disk D. Let Ωn,p1,p2,···,pn = {z = (z1,z2,··· ,zn) ∈ Cn : n: n j=1 |zj|pj < 1}, where 1 ≤ p1 ≤ 2,pj ≥1,j = 2,··· ,n, are real numbers. In this paper, we will prove that Φn,β2,γ2,···,βn,γn(f)(z) =(f(z1),(f(zz11 ))β2(f (z1))γ2z2,··· ,(f(zz11 ))βn(f (z1))γnzn) preserves spirallikeness of type βand order α on Ωn,p1,p2,···,pn.
We give a definition of Bloch space on bounded symmetric domains in arbitrary complex Banach space and prove such function space is a Banach space. The properties such as boundedness, compactness and closed range of composition operators on such Bloch space are studied.
