Fredholm and invertible -tuples of operators. The deformation problem

TLDR: In this paper, the authors considered the class of almost doubly commuting Fredholm pairs with a semi-Fredholm coordinate and obtained a characterization of joint invertibility in terms of a single operator and studied the main examples at length.

Abstract: 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.