Geometric spectral theory for compact operators

We introduce a notion of joint spectrum for a tuple of compact operators on a separable Hilbert space and show that in many situations these operators commute if and only if the joint spectrum consists of countably many, locally finite, complex hyperplanes. In particular, we show that normal matrices (of the same size) $A_1,\cdots,A_n$ commute if and only if the polynomial $\det(z_1A_1+\cdots+z_nA_n+I)$ is completely reducible, that is, it can be factored into a product of linear polynomials.