Cosine operator functions in R^2

Abstract

In this paper, we consider the topic from the theory of cosine operator functions in 2-dimensional real vector space, which is an interplay between functional analysis and matrix theory. For the various cases of a given real matrix A= [α , β; γ , δ] we find out the appropriate cosine operator function C(t)= [a(t), b(t); c(t), d(t)], (t \in R) in a real vector space R² as the solutions of the Cauchy problem C''(t)=AC(t), C(0)=I, C'(0)=0.