Compact Semigroups of Positive Matrices

The spectral theory of compact monothetic semigroups of linear operators examined by Kaashoek and West in [1], [2] together with two block matrix theorems where the blocks are either strictly positive or zero are used to give an exposition of Perron-Frobenius theory of positive matrices. The approach in this paper is based on ideas of Smyth and West developed in [4], [5]. We consider a linear operator T in finite dimensions which has a matrix representation [T] relative to a given basis. Where there is no ambiguity we often write the matrix as T. T ≥ 0 if [T] ij ≥ 0 (∀ i,j) while T > 0 if [T] ij > 0 (∀ i,j). The spectrum and spectral radius of T will be denoted by σ(T) and r(T) respectively. The trace of T (the sum of its eigenvalues) will be written as tr(T), and the peripheral spectrum will be denoted by π(T) = {λ ∈ σ(T); |λ| = r(T)}. The i th row and j th column of T relative to the given basis will be written row i (T) and col j (T) and the diagonal of T will be denoted diag(T). The spectral projection of T relative to π(T) will be written P π. Smyth [5] has introduced a hierarchy of subsets of matrices T ≥ 0. Definitions. (i) T is positive if T > 0; (ii) T is primitive if T k > 0 for some positive integer k; (iii) T is connected if ∀ i,j ∃ a positive integer k such that [T k ] ij > 0; (iv) T is potent if diag(T k) > 0 for some positive integer k; (v) T is zero-free if no row or column is zero; (vi) T has positive spectral radius.