Discrete Approximation of Unbounded Operators and Approximation of their Spectra

Let E be a Banach space over C and let the densely defined closed linear operator A: D(A) ... EQ 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 sa(A) be the approximate point spectrum of A and let se(An) denote the e-pseudospectrum of An. Generalizing our own result, we show that sa(A) ... lim inf se(An)=1n ¥N 4k \ n se(Ak) holds for every e > 0. We deduce that then for every compact set K ... C limn dist(sa(A) 5K, sa(An))=0 provided there exists M> 0 such that ||(l−An)|| [M dist(l, s(An)) holds for every n and every l in the resolvent set r(An) of An. We finally treat the problem under which conditions sa(A) can be approximated from below. More precisely we investigate the problem: Under which assumptions does 4 e > 0 4n ¥N 1k \ n se, a(Ak) ... sa(A) hold where se, a(A) denotes the e-approximate pseudospectrum? © 2001 Elsevier Science