Perturbations of spectra of closed subspaces in Banach spaces

This paper studies the variation of spectra of closed subspaces (i.e., linear relations) in Banach spaces under some perturbations. A concept of commutativity for subspaces in Banach spaces is introduced and its properties are studied. It is shown that boundedness of the inverse of a closed subspace is preserved under relatively bounded and small gap perturbations and perturbation about spectral condition, separately. By using these results, upper semi-continuity of spectra of closed subspaces is obtained and an error estimate of spectra of closed subspaces is given under bounded perturbation. In the special case that the space is a Hilbert space, invariance of self-adjointness of subspaces under relatively bounded and small gap perturbations and the variation of spectrum of a self-adjoint subspace under bounded and Hermitian perturbation are discussed. The results obtained in the present paper generalize the corresponding results for closed operators to closed subspaces and some of which weaken certain assumptions of the related existing results.