  In this paper, we study the unbounded upper triangular operator matrix with
diagonal domain. Some sufficient and necessary conditions are given under which
upper semi-Weyl spectrum (resp. upper semi-Browder spectrum) of such operator
matrix is equal to the union of the upper semi-Weyl spectra (resp. the upper
semi-Browder spectra) of its diagonal entries. As an application, the
corresponding spectral properties of Hamiltonian operator matrix are obtained.
