Samuel multiplicity and the structure of semi-Fredholm operators

Two numerical invariants refining the Fredholm index are introduced for any semi-Fredholm operator in such a way that their difference calculates the Fredholm index. These two invariants are inspired by Samuel multiplicity in commutative algebra, and can be regarded as the stabilized dimension of the kernel and cokernel. A geometric interpretation of these invariants leads naturally to a 4×4 uptriangular matrix model for any semi-Fredholm operator on a separable Hilbert space. This model can be regarded as a refined, local version of the Apostol's 3×3 triangular representation for arbitrary operators. Some classical results, such as Gohberg's punctured neighborhood theorem, can be read off directly from our matrix model. Banach space operators are also considered.     

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.     

The Fredholm index of a pair of commuting operators, II

We first show that an inequality on Hilbert modules, obtained by Douglas and Yan in 1993, is always an equality. This allows us to establish the semi-continuity of the generalized Samuel multiplicities for a pair of commuting operators. Then we discuss the general structure of a Fredholm pair, aiming at developing a model theory. For application we prove that the Samuel additivity formula on Hilbert spaces of holomorphic functions is equivalent to a generalized Gleason problem. As a consequence it follows the additivity of Samuel multiplicity, in its full generality, on the symmetric Fock space. During the course we discover that a variant e^′(⋅) of the classic algebraic Samuel multiplicity might be more suitable for Hilbert modules and can lead to better results.     

3×3上三角算子矩阵的Weyl型定理

设A∈B(H1),B∈B(H2),C∈B(H3)为给定的三个算子,用M(D,E,F)= 表示一个作用在H1(?)H2(?)H3上的3×3算子矩阵．本文首先给出存在算子D∈B(H2,H1),E∈B(H3,H1),F∈B(H3,H2),使得M(D,E,F)为上半Fredholm算子(下半Fredholm算子)的充要条件．同时研究了3×3算子矩阵 M(D,E,F)的Weyl定理,α-Weyl定理,Browder定理和α-Browder定理．     

Helton类算子及单值扩张性质

本文研究了Helton类算子在紧摄动下单值扩张性质的稳定性, 同时研究了2×2上三角算子矩阵在紧摄动下单值扩张性质的稳定性. 利用半Fredholm域的特点, 获得了2×2上三角算子矩阵具有单值扩张性质的稳定性的充分必要条件.     

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.     

An additive formula for Samuel multiplicities on Hilbert spaces of analytic functions

We establish a short exact sequence to relate the germ model of invariant subspaces of a Hilbert space of vector-valued analytic functions and the sheaf model of the corresponding coinvariant subspaces. As a consequence we obtain an additive formula for Samuel multiplicities. As an application, we give a different proof for a formula relating the fibre dimension and the Samuel multiplicity which is first proved in Fang (2005) [11]. The feature of the new proof is that the analytic arguments in Fang (2005) [11] are now subsumed by algebraic machinery.     

上三角无穷维Hamilton算子特征值的代数指标

无穷维Hamilton算子特征函数系是否完备与其代数指标有关,研究了上三角无穷维Hamilton算子特征值的代数指标问题,基于主对角元的特征值和特征向量的某些性质,得到上三角无穷维Hamilton算子的几何重数和代数重数.     

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。