Spectra Of 2×2 Upper Triangular Operator Matrices

The spectral problems of 2x2 upper triangular operator matrix Mc=((?)) acting on a Hilbert space H1(?)H2 are investigated, where A∈B(H1),B∈B(H2) and C∈B(H2,H1). Note that B(X, Y) represents the set of all bounded linear operators from H2 to H1, and is abbreviated to B(X) when Y=X.Firstly, we obtain a necessary and sufficient condition of "a(Mc)=σ(A) Uσ(B) for every C∈B(H2,H1)", in terms of the spectral properties of two diagonal elements A and B in Mc.Also, the analogues for the point spectrum, residual spectrum and continuous spectrum are further presented. In particular, it is shown that the inclusionσΥ(Mc)(?)σΥ(A)∪σΥ(B) for every C∈B(H2, H1) is not correct in general.Secondly, when A∈B(H1),B∈B(H2) are given, the perturbation of the point residual spectrum of 2×2 upper triangular operator matrices Mc= ((?)) is studied. The descriptions of intrinsic and possible point residual spectrum are obtained. Moreover, the connections between the intrinsic point residual spectrum and the union of intrinsic point spectrum, intrinsic residual spectrum are investigated.Finally, based on the consideration on the open problem 3 proposed by professor Du HongKe in [9], we get the conclusions that there exist a operator C0=0(∈B(H2,H1)), such that the intrinsic 1-type point spectrum, intrinsic 1-type residual spectrum and intrinsic continuous spectrum of 2x2 upper triangular operator matrices equal to the corresponding spectrum of M0. Also, these show that "σ*(M0)=σ*(A) Uσ*(B)" is invalid in general, where* is certain spectrum.