Geometric spectral theory for compact operators

Abstract

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