Spectral theory in several variables

1. Recall the spectrum of a bounded linear operator T ∈ B(X) on a Banach space X, more generally a Banach algebra element a ∈ A: 1.1 σ(a) = σA(a) = {λ ∈ C : a− λ 6∈ A−1}, where of course 1.2 a ∈ A−1 ⇐⇒ ∃a′, a′′ ∈ A, a′a = 1 = aa′′ : for example for square matrices A = Cn×n we have the well worn cliche 1.3 σ(a) = σA(a) = {λ ∈ C : det(a− λ) = 0}; for continuous functions A = C(Ω) on compact Hausdorff Ω we have the much more revealing and elementary 1.4 σ(a) = σA(a) = {a(t) : t ∈ Ω}. In a sense spectral theory attempts to reduce general a ∈ A to an appropriate a∧ ∈ C(Ω). The basic properties of ω = σ are 1.5 λ ∈ C =⇒ ω(λ) = {λ} ⊆ C;