Geometric spectral theory for compact operators

Abstract:

For an $n$-tuple $\mathbb {A}=(A_1,\cdots ,A_n)$ of compact operators we define the joint point spectrum of $\mathbb {A}$ to be the set \[ \sigma _p(\mathbb {A})=\{(z_1,\cdots ,z_n)\in \mathbb {C}^n:\ker (I+z_1A_1+\cdots +z_nA_n)\not =(0)\}.\] We prove in several situations that the operators in $\mathbb {A}$ pairwise commute if and only if $\sigma _p(\mathbb {A})$ consists of countably many, locally finite, hyperplanes in $\mathbb C^n$. In particular, we show that if $\mathbb {A}$ is an $n$-tuple of $N\times N$ normal matrices, then these matrices pairwise commute if and only if the polynomial \[ p_{\mathbb {A}}(z_1,\cdots ,z_n)=\det (I+z_1A_1+\cdots +z_nA_n)\] is completely reducible, namely, \[ p_{\mathbb {A}}(z_1,\cdots ,z_n)=\prod _{k=1}^N(1+a_{k1}z_1+\cdots +a_{kn}z_n)\] can be factored into the product of linear polynomials.