Some Aspects of Oscillation and Boundary Problem Theory for Hamiltonian Systems in a $B^ * $-Algebra

Abstract

For linear self-adjoins Hamiltonian differential systems in a $B^ * $-algebra the topics treated include an extension of the well-known generalized polar coordinate transformation for the finite dimensional matrix case, and the derivation of two oscillation criteria for systems that may be written as self-adjoins linear differential equations of the second order. The determination of a Green’s matrix for an incompatible boundary problem involving two-point boundary conditions is discussed, and, in particular, the established results are applied to reduce a certain type of vector boundary problem in a Hilbert space to known results for symmetrizable compact linear transformations in an associated Hilbert space.