A geometric method for eigenvalue problems with low-rank perturbations

We consider the problem of finding the spectrum of an operator taking the form of a low-rank (rank one or two) non-normal perturbation of a well-understood operator, motivated by a number of problems of applied interest which take this form. We use the fact that the system is a low-rank perturbation of a solved problem, together with a simple idea of classical differential geometry (the envelope of a family of curves) to completely analyse the spectrum. We use these techniques to analyse three problems of this form: a model of the oculomotor integrator due to Anastasio & Gad (2007 J. Comput. Neurosci. 22 , 239–254. ( doi:10.1007/s10827-006-0010-x )), a continuum integrator model, and a non-local model of phase separation due to Rubinstein & Sternberg (1992 IMA J. Appl. Math. 48 , 249–264. ( doi:10.1093/imamat/48.3.249 )).

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.