Upper Semi-Weyl and Upper Semi-Browder Spectra of Unbounded Upper Triangular Operator Matrices

In this paper, we study the unbounded upper triangular operator matrix with diagonal domain. Some sufficient and necessary conditions are given under which upper semi-Weyl spectrum (resp. upper semi-Browder spectrum) of such operator matrix is equal to the union of the upper semi-Weyl spectra (resp. the upper semi-Browder spectra) of its diagonal entries. As an application, the corresponding spectral properties of Hamiltonian operator matrix are obtained.

Spectra of 2 x 2 upper triangular operator matrices

The spectra of the 2 x 2 upper triangular operator matrix MC = (A C 0 B ) acting on a Hilbert space H1 ? H2 are investigated. We obtain a necessary and sufficient condition of ?(MC) = ?(A)??(B) for every C ? B(H2,H1), in terms of the spectral properties of two diagonal elements A and B of MC. Also, the analogues for the point spectrum, residual spectrum and continuous spectrum are further presented. Moveover, we construct some examples illustrating our main results. In particular, it is shown that the inclusion ?r(MC) ? ?r(A) ? ?r(B) for every C ? B(H2,H1) is not correct in general. Note that ?(T) (resp. ?r(T)) denotes the spectrum (resp. residual spectrum) of an operator T, and B(H2,H1) is the set of all bounded linear operators from H2 to H1.

The residual spectrum and the continuous spectrum of upper triangular operator matrices

Let H and K be separable infinite dimensional Hilbert spaces. We denote by MC the 2x2 upper triangular operator matrix acting on H ? K of the form MC = (A C/0 B ). For given operators A ? B(H) and B ? B(K), the sets C?B?(K,H) ?r(MC) and C?B?(K,H) ?c(MC) are characterized, where ?r(?) and ?c(?) denote the residual spectrum and the continuous spectrum, respectively