The generalized Roper-Suffridge extension operatorΦ(f) on the bounded complete Rein- hardt domainΩin C^n with n≥2 is defined byΦ_(n,β_2,γ_2,...,β_n,γ_n)~r(f)(z)=rf((z_1)/r),((rf(z_1)/r)/(z_1))^(β2)(f′((z_1)/r)^(γ2)z_2,...,((rf(z_1)/r)/(z_1))^(β_n)(f′((z_1)/r))^(γ_n)z_n) for (z_1,z_2,...,z_n)∈Ω,where r=r(Ω) = sup{|z_1|:(z_1,z_2,...,z_n)∈Ω},0≤γ_j≤1-β_j,0≤β_j≤1, and we choose the branch of the power functions such that ((f(z_1))/(z_1))^(β_j)|_(z_1=0)=1 and (f′(z_1))^(γ_j)|_(z_1=0)= 1,j=2,...,n.In this paper,we prove that the operatorΦ_(n,β_2,γ_2,...,β_n,γ_n)~r(f) is from the subset of S_α~*(U) to Sα~*(Ω)(0≤α<1) onΩand the operatorΦ_(n,β_2,γ_2,...,β_n,γ_n)~r(f) preserves the starlikeness of order a or the spirallikeness of typeβon D_p for some suitable constantsβ_j,γ_j,p_j,where D_p= {(z_1,z_2,...,z_n)∈C^n:∑_(j=1)~n|z_j|^(p_j)<1}(p_j>0,j=1,2...,n),U is the unit disc in the complex plane C,and S_α~* (Ω) is the class of all normalized starlike mappings of orderαonΩ.We also obtain thatΦ_(n,β_2,γ_2,...,β_n,γ_n)~r(f)∈S_α~*(D_p) if and only if f∈S_α~*(U) for 0≤α<1 and some suitable constantsβ_j,γ_j,p_j.
Yu-can ZHU~(1+) Ming-sheng LIU~2 ~1 Department of Mathematics,Fuzhou University,Fuzhou 350002,China
In this article, the generalized Roper-Suffridge extension operator in Banach spaces for locally biholomorphic mappings is introduced. It is proved that this operator preserves the starlikeness on some domains in Banach spaces but does not preserves convexity for some cases. Moreover, the growth theorem, covering theorem, and the radius of starlikeness are discussed. Some results of Roper and Suffridge, Gong and Liu, Graham et al in C^n are extended to Hilbert spaces or Banach spaces.
In this paper, we introduce the pre-frame operator Q for the g-frame in a complex Hilbert space, which will play a key role in studying g-frames and g-Riesz bases etc. Using the pre-frame operator Q, we give some necessary and sufficient conditions for a g-Bessel sequence, a g-frame, and a g-Riesz basis in a complex Hilbert space, which have properties similar to those of the Bessel sequence, frame, and Riesz basis respectively. We also obtain the relation between a g-frame and a g-Riesz basis, and the relation of bounds between a g-frame and a g-Riesz basis. Lastly, we consider the stability of a g-frame or a g-Riesz basis for a Hilbert space under perturbation.
