4 Notes on normed algebras , 4

Let A be a finite-dimensional algebra over the complex numbers with nonzero identity element e. If x ∈ A, then the resolvent set associated to x is the set ρ(x) of complex numbers λ such that λ e − x is invertible, and the spectrum of x is the set σ(x) of complex numbers λ such that λ e − x is not invertible. For instance, if V is a finite-dimensional vector space over the complex numbers of positive dimension, L(V ) is the algebra of linear operators on V , and T is a linear operator on V , then a complex number λ lies in the spectrum of T if and only if λ I − T has a nontrivial kernel. This is equivalent to saying that there is a nonzero vector v ∈ V such that T (v) = λ v, which is to say that v is a nonzero eigenvector for T with eigenvalue λ. Let p(z) be a polynomial on the complex numbers, which can be written explicitly as p(z) = cm z m + cm−1 z m−1 + · · ·+ c0 (1)