半Fredholm算子的一个注记

本文研究了(上,下)半Fredholm算子的结构问题.利用经典的算子理论方法,由闭不变子空间诱导的两个映射,获得了(上,下)半Fredholm算子的多种新的等价刻画,推广了(上,下)半Fredholm算子的已有结果.     

一类缺项算子矩阵的半Fredholm补

设H,K为可分Hilbert空间,A∈B(H),B∈B(H,K)和D∈B(K)是给定的有界线性算子,定义缺项算子矩阵N_C=(ABCD).得到存在C∈B(K,H)使得N_C是上半Fredholm算子(下半Fredholm算子,Fredholm算子)的条件.     

保持固定零度的半Fredholm算子的映射

设$X$是一个无限维复Banach空间, $B(X)$是$X$上所有有界线性算子的代数. 给定整数$n\ge 1$, 我们刻画了$B(X)$上的所有双向保持零度等于$n$的半Fredholm算子的差的双射,并建立了映射的结构.     

关于Fredholm算子序列

本文引入了Fredholm算子序列,并讨论了它的一些性质,得到了许多类似单个Fredholm算子的一些特征和性质。     

正规算子和亚正规算子的一些特征

In this paper it is proved that if T is a bounded linear operator on a Hilbert space H and λ(?)c1(W(T)), where cl(W(T)) is the closure of W(T)={(Tx, x); x∈H, ‖x‖ =1}, then T is normal iff U_λ=(T-λ)^*-1 (T-λ) is hyponormal and T is hyponormal iff U_λ=(T-λ)^*-1 (T-λ) is normaliod.     

Hilbert空间子空间的维数和半Fredholm算子的指标

对Hilbert空间H中的任意子空间,引入了一种广义维数并使其全序化.发现了可列个无限维数:∞~*<∞ⁿ<∞^m,其中n,m为整数且n>m.由此定义了半Fredholm算子SF(H)的广义指标.证明任意纯半Fredholm算子A∈SF₊(H)(SF_-(H))有Ind_gA=∞^*(-∞^*).这里的广义维数和指标是几何与拓扑的概念,像有限维数和Fredholm指标一样具有某些分析演算功能.作为例,证明了:对任意等距算子V₁,V₂,它们在H上左可逆算子B_i^x(H)中道路连通当且仅当Ind_gV₁=Ind_gV₂.从而推出,(?) A,B∈SF₊(H)(SF_-(H)),它们为道路连通的充要条件为Ind_gA=Ind_gB.其余有关SF(H)的结果,如指标经紧和小扰动的稳定性,Ind_g:SF(H)→ZU{-∞^*,∞^*}连续皆成立.     

具有单值扩张性的上三角算子矩阵半Fredholm性及其应用

本文研究了具有单值扩张性的上三角算子矩阵半Fredholm性,利用单值扩张性与分块算子升标、降标、零维、亏维之间的联系,得到了具有单值扩张性的上三角算子矩阵半Fredholm性刻画,给出了用对角算子刻画上三角算子矩阵半Fredholm性的条件,并研究了算子矩阵半Fredholm性的扰动问题;此外,利用所得结果研究了上三角算子矩阵的谱、本质谱和Browder谱,同时进一步考虑了Browder定理,a-Browder定理,Weyl定理,a-Weyl定理,从局部谱的角度揭示了定理之间的联系,得到了定理成立的新条件并举例验证.     

空间上加权复合算子的Fredholm性

This paper characterizes bounded, Fredholm weighted composition operators on Hardy space. As an application, the Fredholm index of such operators is obtained.     

On the Cellular Indecomposable Property of Semi-Fredholm Operators

The authors prove that an operator with the cellular indecomposable property has no singular points in the semi-Fredholm domain, by applying the 4 × 4 matrix model of semi-Fredholm operators due to Fang in 2004. This result fills a gap in the result of Olin and Thomson in 1984.     

S.G理论在研究Semi-Fredholm算子时的应用

本文研究了具有有限升标（降标）的半Fredholm算子，证明了具有有限 升标（降标）的上半Fredholm算子在其摄动类中交换元的摄动下仍具有同样性质，对于下半Fredholm算子有同样结论。从而改进了[1,2]中的主要结果。同时，我们证明了被摄动算子集合扩大（相对于[1]而言）而摄动仍为紧摄动时较[1]中更强的结果。     

Fredholm and Semi-Fredholm Composition Operators on the Small Bloch-type Spaces

We characterize the Fredholm composition operators on the small Bloch-type spaces ${\mathcal{B}^{\alpha}}$ , with 0 < α < 1, and give necessary and sufficient conditions for their semi-Fredholmness. For nontrivial composition operators the semi-Fredholmness is equivalent to the operator having a closed range, i.e. being bounded below. These properties have been characterized when the composition operator acts on the Bloch-type spaces ${\mathcal{B}^{\alpha}, \alpha \ge1}$ . We extend some of those results, and give a few characterizations, for the leftover case of small Bloch-type spaces.     

A Characterization of BMO via Commutators for Some Operators

In this paper we give a characterization of BMO via the boundedness of Tb and Ia,b, the commutator of a singular integral operator and a fractional integral operator respectively,on generalized Morrey spaces Lp,φ(R).Moreover, we prove that the fractional maximal operator Ma and the fractional integral operator Ia are bounded operators from Lp,φ to Lq,φ(R) when 1/q=1/p-a/(n-λ).     

On Some Zero-Increasing Operators

We characterize the set of linear operators of the form $$Tx^n = x^n + \sum\limits_{k = 0}^{n - 1} {b_{n,k} x^k ,} n = 0,1, \ldots ,$$ which satisfy Z(T _p ) ≧ Z(p) for every polynomial p, where Z counts the number of zeros on all of R. We also consider the analogous question on [0,∞)].     

Characterization of Non-Smooth Pseudodifferential Operators

Smooth pseudodifferential operators on \(\mathbb {R}^{n}\) can be characterized by their mapping properties between \(L^p-\)Sobolev spaces due to Beals and Ueberberg. In applications such a characterization would also be useful in the non-smooth case, for example to show the regularity of solutions of a partial differential equation. Therefore, we will show that every linear operator P, which satisfies some specific continuity assumptions, is a non-smooth pseudodifferential operator of the symbol-class \(C^{\tau } S^m_{1,0}(\mathbb {R}^n \times \mathbb {R}^n)\). The main new difficulties are the limited mapping properties of pseudodifferential operators with non-smooth symbols.