On the rank of the matrix radius of the limiting set for a singular linear Hamiltonian system

This paper is concerned with the rank of the matrix radius of the limiting set for a singular Hamiltonian system with one singular end point. The exact relationship between the rank of the matrix radius and the number of square integrable solutions is obtained and then the defect index of the corresponding minimal operator can be represented in terms of the rank of the matrix radius. So two results obtained by Allan M. Krall [SIAM J. Math. Anal. 20 (1989) 664] are improved. In addition, it is discussed that the rank of the matrix radius is independent of the spectral parameter and a certain matrix. Especially, the classification of singular linear Hamiltonian systems is present and several sufficient and necessary conditions for the limit point and limit circle cases are established.