紧Riemann面上的复合算子

In the present paper, the characterization of invertible composition oper-ators on compact Riemann surfaces is obtained, which is very different from that in the case of Hardy or Bergman spaces on a disk of the complex plane C.     

Operators on Differential Form Spaces for Riemann Surfaces

In the present paper, a problem of Ioana Mihaila is negatively answered on the invertibility of composition operators on Riemann surfaces, and it is proved that the composition operator Cp is Predholm if and only if it is invertible if and only if p is invertible for some special cases. In addition, the Toeplitz operators on ∧1 2, a（M） for Riemann surface M are defined and some properties of these operators are discussed.     

解析函数的Banach空间上之复合算子

本文研究了一类解析函数的Ｂａｎａｃｈ空间Ｘ上之复合算子，这类空间包含了Ｂｌｏｃｈ空间，并且可看作Ｂｅｒｇｍａｎ空间Ｌ１ａ（Ｄ）中具有原子分解的解析函数的对偶空间．我们刻划了这类空间上紧复合算子及Ｆｒｅｄｈｏｌｍ复合算子的特征，此外，还研究了具有闭值域的复合算子．     

Fredholm Operators and Riesz Theory for Polynomially Compact Operators

In this article, we give a characterization of a class of bounded Fredholm operators on a Banach space which is developed to present some general existence results of the operators equation of the second kind. The obtained results are used to describe the Riesz–Schauder theory of compact operators in the more general setting of polynomially compact operators.     

Fredholm composition operators on algebras of analytic functions on Banach spaces

We prove that Fredholm composition operators acting on the uniform algebra H^∞(BE) of bounded analytic functions on the open unit ball of a complex Banach space E with the approximation property are invertible and arise from analytic automorphisms of the ball.     

Gromov Hyperbolicity of Riemann Surfaces

We study the hyperbolicity in the Gromov sense of Riemann surfaces. We deduce the hyperbolicity of a surface from the hyperbolicity of its ＂building block components＂. We also prove the equivalence between the hyperbolicity of a Riemann surface and the hyperbolicity of some graph associated with it. These results clarify how the decomposition of a Riemann surface into Y-pieces and funnels affects the hyperbolicity of the surface. The results simplify the topology of the surface and allow us to obtain global results from local information.     

Bergman空间上Toeplitz算子的局部及Fredholm性质

本文讨论了Ｂｅｒｇｍａｎ空间上Ｔｏｅｐｌｉｔｚ算子的局部谱的关系．利用局部化原理给出了Ｔｏｅｐｌｉｔｚ算子拟换位子紧性的一个充分条件．给出了一类符号的Ｔｏｅｐｌｉｔｚ算子是Ｆｒｅｄｈｏｌｍ算子的充分条件．     

Composition operators on a Banach space of analytic functions

In this paper, we study composition operators on a Banach space of analytic functions, denoted byX, which includes the Bloch space. This space arises naturally as the dual space of analytic functions in the Bergman spaceL _α ¹ (D) which admit an atomic decomposition. We characterize the functions which induce compact composition operators and those which induce Fredholm operatorson this space. We also investigate when a composition operator has a closed range. Supported by NNSFC No.19671036     

Fredholm and Propernes Properties of Quasilinear Elliptic Operators on RN

We discuss the Fredholm and properness properties of second-order quasilinear elliptic operators viewed as mappings from W^2,p( R ^N) to L^p( R ^N) with N < p < ∞. The unboundedness of the domain makes the standard Sobolev embedding theorems inadequate to investigate such issues. Instead, we develop several new tools and methods to obtain fairly simple necessary and suffcient conditions for such operators to be Fredholm with a given index and to be proper on the closed bounded subsets of W^2,p( R ^N). It is noteworthy that the translation invariance of the domain, well-known to be responsible for the lack of compactness in the Sobolev embedding theorems, is taken advantage of to establish results in the opposite direction and is indeed crucial to the proof of the properness criteria. The limitation to second-order and scalar equations chosen in our exposition is relatively unimportant, as none of the arguments involved here relies upon either of these assumptions. Generalizations to higher order equations or to systems are thus clearly possible with a variableamount of extra work. Various applications, notably but not limited, to global bifurcation problems, are described elsewhere.     

Dirichlet空间上的循环复合算子

该文主要证明了在Dirichlet空间上由复合算子{C\-φ:φ∈Aut(D)}生成的代数为循环算子代数;同时对任意的解析映射φ:D→D,C\-φ都不可能为超循环算子给出了证明.     

Riemann曲面之间的极值拟共形映射

设Ｓ和Ｒ是两个以单位圆为万有覆盖的Ｒｉｅｍａｎｎ曲面,ｆ：Ｓ→Ｒ为拟共形同胚,类似于ＫＳｔｒｅｂｅｌ的方法。我们引ｉｅｍａｎｎ嗣面Ｓ上的点ｐ０关于模边同伦类「ｆ０」的可变性集合Ｖ「ｆ０」「ｐ０」的概念并且证明可变性集合是Ｒ的一个紧的连通子集。     

Hardy空间上的有界复合算子

本文讨论了由解析映射φ：Ｂｎ→Ｂｍ诱导复合算子的有界性；特别地，研究了由保测内映射诱导复合算子的性质．作为推论，得到了Ｗ．Ｒｕｄｉｎ［１０］中一个公开问题的解答．     

Dirac Operators on Hermitian Spin Surfaces

We prove the conformal invariance of the dimension of thekernel of any of the self-adjoint Dirac operators associated to thecanonical Hermitian connections on Hermitian spin surface. In the caseof a surface of nonnegative conformal scalar curvature we estimate thefirst eigenvalue of the self-adjoint Dirac operator associated to theChern connection and list the surfaces on which its kernel isnontrivial.     

HardyOrlicz空间之间的加权复合算

该文研究了复平面中单位圆盘上不同Hardy-Orlicz空间之间的加权复合算子,利用Carleson测度不等式给出了有界或紧的加权复合算子ωC_φ:N_p→N_q的特征. 作为推论得到了加权复合算子ωC_φ:N_p→N_q有界(或紧)的充分必要条件是ωC_φ:H_p→H_q是有界(或紧)的. 此外,还给出了Hardy-Orlicz空间上可逆及Fredholm复合算子的特征.     

Generalized Weighted Composition Operators on Weighted Bergman Spaces

The boundedness and compactness of the generalized weighted composition operator on weighted Bergman spaces are discussed.     

The Hyperbolic Geometric Flow on Riemann Surfaces

In this paper the authors study the hyperbolic geometric flow on Riemann surfaces. This new nonlinear geometric evolution equation was recently introduced by the first two authors, motivated by Einstein equation and Hamilton's Ricci flow. We prove that, for any given initial metric on ?² in certain class of metrics, one can always choose suitable initial velocity symmetric tensor such that the solution exists for all time, and the scalar curvature corresponding to the solution metric g _ij keeps uniformly bounded for all time; moreover, if the initial velocity tensor is suitably “large", then the solution metric g _ij converges to the flat metric at an algebraic rate. If the initial velocity tensor does not satisfy the condition, then the solution blows up at a finite time, and the scalar curvature R(t, x) goes to positive infinity as (t, x) tends to the blowup points, and a flow with surgery has to be considered. The authors attempt to show that, comparing to Ricci flow, the hyperbolic geometric flow has the following advantage: the surgery technique may be replaced by choosing suitable initial velocity tensor. Some geometric properties of hyperbolic geometric flow on general open and closed Riemann surfaces are also discussed.     

Bergman空间上的复合算子的范数

本文研究了Bergman空间上的复合算子的范数与再生核的关系,证明了紧复合算子C的范数‖C‖=sup{‖C*kw‖:w∈D}的充要条件是(0)=0或是仿射映射,即(z)=sz+t,s,t是满足|s|+|t|<1的常数,其中kw为Bergman空间的规范再生核, C*是C的共轭算子.     

单位球上不同Hardy空间之间的复合算子

本文研究了ＣＮ中单位球Ｂ上不同Ｈａｒｄｙ空间之间的复合算子,引入了η－Ｃａｒｌｅｓｏｎ测度,利用它作为工具给出了有界的或紧的复合算子Ｃ■： Ｈｐ（Ｂ） → Ｈｑ（Ｂ）的特征．     

Computing the Fredholm index of Toeplitz operators with continuous symbols

We show how to compute the Fredholm index of a Toeplitz operator with a continuous symbol constructed from any subnormal operator with compact self-commutator. We also show that the essential spectral pictures of such Toeplitz operators can be prescribed arbitrarily.