Fredholm and invertible -tuples of operators. The deformation problem
Fredholm and invertible -tuples of operators. The deformation problem
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DOI：
10.1090/s0002-9947-1981-0613789-6
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发表时间：
1981
影响因子：
1.3
通讯作者：
R. Curto
中科院分区：
文献类型：
--
作者：
R. Curto
利用J. L. Taylor的联合谱定义，研究了Hilbert空间上算子的Fredholm和可逆-元组。我们给出了“多变量”理论的基础，包括阿特金森定理的一个自然推广和一个表现良好的指标。我们得到了单算子的联合可逆性的一个表征，并详细地研究了主要的例子。然后考虑一类具有半Fredholm坐标的几乎双重交换Fredholm对的变形问题并求解它。1. 介绍。1。设T是巴拿赫空间%上的一个（有界线性）算子。如果在%上存在一个算子S，使得TS = ST = 1%，即9C上的恒等算子T是可逆的。根据开放映射定理，这等价于ker T =(0)和R(T) = T = %的范围。最后一个公式不依赖于T的逆的存在性，而是依赖于算子T的作用。当T被交换算子的元组取代时，存在几种非奇异性的定义。J. L. Taylor（19）通过考虑与“-元组”相关的Koszul复合体，得到了一个反映操作者行为的复合体。2. 本文在泰勒工作的基础上发展了一般的“多变量”理论，并研究了Hilbert空间上算子的可交换和几乎可交换（=可交换模- 0紧）-元组。我们用单个算子的可逆性得到了联合可逆性的一个表征，这对我们的方法是至关重要的。由此，我们得到了一些推论，这些推论很好地推广了“一个变量”下已知的初等结果。同时，所提到的性质允许我们在Fred- holm ‘ -元组（在Calkin代数中可逆的几乎可交换’ -元组）的类上定义一个连续的、紧摄动下不变的整值指标。这个索引扩展了Fredholm算子的经典索引。我们证明了在跟踪类中具有所有交换子的基本正常算子的几乎交换元组具有索引0(> 2)，并且证明了Atkinson定理的一个自然推广适用于元组。
Using J. L. Taylor's definition of joint spectrum, we study Fredholm and invertible «-tuples of operators on a Hilbert space. We give the foundations for a "several variables" theory, including a natural generalization of Atkinson's theorem and an index which well behaves. We obtain a characterization of joint invertibility in terms of a single operator and study the main examples at length. We then consider the deformation problem and solve it for the class of almost doubly commuting Fredholm pairs with a semi-Fredholm coordinate. 1. Introduction. 1. Let T be a (bounded linear) operator on a Banach space %. T is said to be invertible if there exists an operator S on % such that TS = ST = 1%, the identity operator on 9C. By the Open Mapping Theorem, this is equivalent to ker T = (0) and R(T) = range of T = %. The last formulation does not rely upon the existence of an inverse for T, but rather on the action of the operator T. When T is replaced by an «-tuple of commuting operators, several definitions of nonsingular- ity exist. J. L. Taylor (19) has obtained one which reflects the actions of the operators, by considering the Koszul complex associated with the «-tuple. 2. In this paper we develop a general "several variables" theory on the basis of Taylor's work and study commuting and almost commuting (= commuting mod- ulo the compacts) «-tuples of operators on a Hilbert space %. We obtain a characterization of joint invertibility in terms of the invertibility of a single operator, which is essential for our approach. From that we get a number of corollaries which generalize nicely the known elementary results in "one variable". At the same time, the referred characterization allows us to define a continuous, invariant under compact perturbations, integer-valued index on the class of Fred- holm «-tuples (those almost commuting «-tuples which are invertible in the Calkin algebra). This index extends the classical one for Fredholm operators. We prove that an almost commuting «-tuple of essentially normal operators with all commu- tators in trace class has index zero (« > 2) and that a natural generalization of Atkinson's theorem holds for «-tuples.