Possible Spectrums Of 3×3 Upper Triangular Operator Matrices

Let H₁,H₂ and H₃ be complex separable infinite dimensional Hilbert spaces. Denoteβ(H_i),β(H_i, H_j) (i, j = 1,2,3) the Banach spaces of all the bounded linear operators on H_i and from H_i into H_j, respectively. For given operators A∈β(H₁),B∈β(H₂) and C∈β(H₃), let M_(d,e,f) denote the 3×3 upper triangular operator matrixThere are four chapter in this article, and main concept as follow:In chapter 1, the background, the developing general situation of operator matrices completion problems and the main results obtained in this article are described.In chapter 2, the possible residual spectrum of 3x3 upper triangular operator matrix M_(d,e,f) are studied.In chapter 3, the possible continuous spectrum of 3x3 upper triangular operator matrix M_(d,e,f) are studied.In chapter 4, the possible point spectrum and the possible spectrum for 3×3 upper triangular operator matrix M_(d,e,f) are discussed firstly. Secondly, we point out the possible spectrum for M_(M,E,F) is equals to the union set of the possible residual spectrum, the possible continuous spectrum and the possible point spectrum. Finally, several examples are presented to verify the correctness of the result.