The boundary spectrum of linear operators in finite-dimensional spaces
Yuri I. Lyubich · Banach Center Publications · 1997
1. The basic concepts. Let X be an n-dimensional real or complex vector space, n < ∞. Let A : X → X be a linear operator. It is called power bounded if the semigroup Π(A) = {A}0 of its natural powers is bounded. (This property can be defined in terms of any norm: sup{‖A‖ : k ∈ N} < ∞. The choice of the norm does not matter.) An operator A is called double power bounded if A is invertible and A,A are both power bounded. (Thus, sup{‖A‖ : k ∈ Z} < ∞.) We only consider the power bounded operators. Since in this case the spectral radius r(A) does not exceed 1, the spectrum σ(A) consists of two parts, the boundary (or peripheral) spectrum