Compact Semigroups of Positive Matrices

T. T. West
2003 Mathematical Proceedings of the Royal Irish Academy  
The spectral theory of matrices generating compact semigroups is combined with results on block positive matrices to obtain the Frobenius representation for connected nonnegative matrices. Introduction 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, is used to give an exposition of Perron-Frobenius theory of positive matrices. This
more » ... e matrices. This approach is based on ideas of Smyth and West developed in We consider a linear operator T with finite dimensions that has a matrix representation [T ] relative to a given basis. Where there is no ambiguity we often write the matrix as . The spectrum and spectral radius of T will be denoted by s(T ) and r(T ), respectively. The trace of T (the sum of its diagonal entries) will be written as tr(T ), and the peripheral spectrum will be denoted by p(T )={l 2 s(T ); |l|=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 p(T ) will be written P p . Smyth [5] has introduced a hierarchy of subsets of matrices Ti0. Definitions. The following attributes are appropriate to any matrix Ti0: for some positive integer k; (v) T is zero-free if no row or column is zero ; (vi) T has positive spectral radius. Remarks. The above sets are strictly ordered by inclusion. T is connected, if and only if there exists a positive integer p such that T+T 2 +...+T p >0. It is also connected if no basis permutation results in a block representation
doi:10.3318/pria.2003.103.2.143 fatcat:y2wtenfajfbuncxahofb6ft5ii