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
In this article, the authors study the growth of certain second order linear differential equation f″+A(z)f′+B(z)f=0 and give precise estimates for the hyperorder of solutions of infinite order. Under similar conditions, higher order differential equations will be considered.
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.