The Spectral Equality Of 2×2 Upper Triangular Operator Matrices With Unbounded Entries

In this dissertation,we study the necessary and sufficient conditions for 2×2 un-bounded upper triangular operator matrices (?) in infinite dimensional complex separable Hilbert spaces to be surjective,bounded below and invertible.Moreover,some necessary and sufficient conditions under which ?_*(T)=?_*(A)??_*(D)holds are obtained,?_*={?_?,?_ap,?}.These results generalize the sufficient conditions given by Du,Han and Barraa in the case of bounded operator matrices.As applications,the necessary and sufficient conditions for invertible and spectral equality of the diagonally dominant upper triangular infinite dimensional Hamilton operators are given,and examples are given to verify the validity.