Possible Spectrums of 3×3 Upper Triangular Operator Matrices

Let H1,H2 and H3 be infinite dimensional separable complex Hilbert spaces.We denote by M(D,E,F) a 3×3 upper triangular operator matrix acting on H1⊕ H2⊕ H3 of the form M(D,E,F)=.For given A∈B(H1),B∈B(H2) and C∈B(H3),the sets ∪D,E,F σp(M(D,E,F)),∪D,E,F σr(M(D,E,F)),∪D,E,F σc(M(D,E,F)) and ∪D,E,F σ(M(D,E,F)) are characterized,where D∈B(H2,H1),E∈B(H3,H1),F∈B(H3,H2) and σ(·),σp(·),σr(·),σc(·) denote the spectrum,the point spectrum,the residual spectrum and the continuous spectrum,respectively.