<b>Perturbation of Weyl’s Theorems for Unbounded  Upper Triangular Operator Matrices</b>

Let be an upper triangular operator matrix which is unbounded and defined on , where is infinite dimensional Hilbert space. This paper is concerned with new spectral properties which defined to other bounded operators. Some sufficient and necessary conditions are given in which these properties are equivalent. We further investigate the relations among Weyl’s type theorems and Brodwe’s theorems for this type of operator under some conditions. As an application the paper define the plate pending problem equation with henge end, fixed end and free end, after transform it to Hamitonian matrix then calculate the spectrum sets for this matrix which leads to if A has eigenvalues of finite multiplicity, so is M. Inaddition if has finite ascent this implies that the Hamiltonian operator M has finite ascent