  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.
