Fredholm and invertible 𝑛-tuples of operators. The deformation problem

Abstract

Using J. L. Taylor’s definition of joint spectrum, we study Fredholm and invertible n n -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.