On the symplectic self-adjointness and residual spectral emptiness of upper triangular Hamiltonian operator matrices

Abstract

This paper deals with the symplectic self-adjointness and residual spectral emptiness of upper triangular Hamiltonian operator matrices \(H=\left( {\begin{matrix}A&{}B\\ 0&{}-A^*\end{matrix}}\right) \). First, for symplectic self-adjoint Hamiltonian operator H, based on detailed classification of point spectrum \(\sigma _p(H)\) and residual spectrum \(\sigma _r(H)\), the symmetry about imaginary axis is given between \(\sigma _p(H)\), \(\sigma _r(H)\), deficiency spectrum \(\sigma _{\delta }(H)\), compression spectrum \(\sigma _\mathrm{{com}}(H)\) and approximate point spectrum \(\sigma _\mathrm{{app}}(H)\). Second, by means of the spectral symmetry, the sufficient and necessary conditions are given for \(\sigma _r(H)=\varnothing \), \(\sigma _{r_1}(H)=\varnothing \) and \(\sigma _{r_2}(H)=\varnothing \), respectively. Then, for \(H=\left( {\begin{matrix}A&{}B\\ 0&{}-A^*\end{matrix}}\right) \), it is proved that H is symplectic self-adjoint, if H is defined with diagonal domain \({\mathcal {D}}(H)={\mathcal {D}}(A)\oplus {\mathcal {D}}(A^*)\). Finally, for \(H=\left( {\begin{matrix}A&{}B\\ 0&{}-A^*\end{matrix}}\right) \) defined with diagonal domain, using the space decomposition, the sufficient and necessary conditions for \(\sigma _r(H)=\varnothing \) and \(\sigma _{r_1}(H)=\varnothing \) are described in detail, respectively, by line operator, null space, and range of inner elements.