Relatively Bounded and Relatively Compact Perturbations for Limit Circle Hamiltonian Systems

This paper is concerned with relatively bounded and relatively compact perturbations of limit circle Hamiltonian systems of arbitrary order which may be formally non-symmetric. In this paper, a relationship between the relative boundedness and relative compactness of an operator is obtained, and a sufficient and necessity condition is given for a relatively compact operator V with respect to an operator T to be relatively compact with respect to a finite-dimensional closed extension of T. Furthermore, properties of the limit circle Hamiltonian systems are derived, the regularity field of the minimal operator corresponding to the limit circle Hamiltonian system is given, and then it is proved that the relative boundedness and relative compactness of a class of multiplication operators with respect to the maximal operator corresponding to the limit circle Hamiltonian system are equivalent.