Positivity (2017) 21:755–786
Spectral theory in ordered Banach algebras
· Heinrich Raubenheimer
Received: 1 October 2015 / Accepted: 9 August 2016 / Published online: 28 September 2016
© Springer International Publishing 2016
Abstract We give a survey of the development of the spectral theory in ordered
Banach algebras; from its roots in operator theory to the modern abstract context.
Keywords Ordered Banach algebra · Spectral theory
Mathematics Subject Classiﬁcation 46H05 · 47A10 · 47B65
Let T be an n × n matrix with complex entries. One can view T as an operator of C
in the normal way. It was discovered around the turn of the previous century
that the spectrum of an n × n matrix with positive entries has certain special features.
The ﬁrst result was by Perron.
Theorem 1.1 () Let T =[t
] be an n × n matrix with t
> 0 for all i and j.
1. T has strictly positive spectral radius r (T ).
2. r(T ) is a simple eigenvalue of T with strictly positive eigenvector.
3. T is primitive, i.e. r(T ) is the unique eigenvalue on the spectral circle |λ|=r(T ).
Department of Mathematical Sciences, Stellenbosch University, Private Bag X1, Matieland
7602, South Africa
Department of Mathematics, University of Johannesburg, P. O. Box 524, Auckland Park 2006,