Compact Operators and Kernel Bundles

As a bridge between matrices (or finite rank operators) and linear operators on an infinite dimensional Hilbert space \({\mathcal H}\) , compact operators play an important role in operator theory and beyond. Indeed, the study of their invariant subspaces has led us to some of the deepest theorems, and the Calkin algebra \(B({\mathcal H})/K({\mathcal H})\) is the ground for the definition of Fredholm operators which are fundamental to spectral theory, index theory, and noncommutative geometry. It is therefore of great interest to gain a somewhat in-depth understanding of the projective spectrum of compact operators.