Quasi self-adjoint extensions of a class of formally non-self-adjoint discrete Hamiltonian systems

Abstract

This paper is concerned with the characterizations of quasi self-adjoint extensions of a class of formally non-self-adjoint discrete Hamiltonian systems. Some properties of the solutions and the characterization of the minimal linear relations of the non-self-adjoint systems are obtained. A bijective projection between all the quasi self-adjoint extensions of non-self-adjoint systems and all the self-adjoint extensions of the self-adjoint systems generated by the non-self-adjoint Hamiltonian systems is established in the general case. When the system is in the limit point case and $\I=[a,\infty)$, a complete characterization of all the quasi self-adjoint extensions is obtained by a subspace $Q\subset \C^{2n}$ with $\dim Q=n$ in terms of boundary conditions.