Discrete Approximation of Unbounded Operators and Approximation of their Spectra Original Research Article

Let E be a Banach space over C and let the densely defined closed linear operator A: D(A)⊂E→E be discretely approximated by the sequence ((An, D(An)))n∈N of operators An where each An is densely defined in the Banach space Fn. Let σa(A) be the approximate point spectrum of A and let σε(An) denote the ε-pseudospectrum of An. Generalizing our own result, we show that σa(A)⊂lim inf σε(An)=∪n∈N ∩k⩾n σε(Ak) holds for every ε>0. We deduce that then for every compact set K⊂C limn dist(σa(A)∩K, σa(An))=0 provided there exists M>0 such that ‖(λ−An)−1‖⩽M dist(λ, σ(An))−1 holds for every n and every λ in the resolvent set ρ(An) of An. We finally treat the problem under which conditions σa(A) can be approximated from below. More precisely we investigate the problem: Under which assumptions does ∩ε>0 ∩n∈N ∪k⩾n σε, a(Ak)⊂σa(A) hold where σε, a(A) denotes the ε-approximate pseudospectrum?