Closedness Of Ranges Of3×3Upper-triangular Operator Matrices

Let H1, W2and H3be separable complex Hilbert spaces. We denote by M a3x3upper-triangular operator matrice acting on H1(?)H2(?)H3.Let A∈B(H1),B∈B(H2), C∈B(H3) and E∈B(H2,H1) be given operators. Firstly, the closedness of the range R(M) is described by using the ranges and null spaces of A, B and C, and its Moore-Penrose possible spectrum and inherent spectrum are described as well.Moreover, the four kinds of point spectra of the2×2upper-triangular operator are described, which provide a foundation for studying the point spectra of3x3upper-triangular operator M.