Geometric spectral theory for compact operators

For an n-tuple A = (A1,··· ,An) of compact operators we define the joint point spectrum of A to be the set p(A) = {(z1,··· ,zn) ∈ C n : ker(I + z1A1 + ··· + znAn) 6 (0)}. We prove in several situations that the operators in A pairwise commute if and only ifp(A) consists of countably many, locally finite, hyper- planes in C n . In particular, we show that if A is an n-tuple of N × N normal matrices, then these matrices pairwise commute if and only if the polynomial