Discrete approximation of unbounded operators and approximation of their spectra

[摘要] Let E be a Banach space over C and let the densely defined closed linear operator A: D(A) subset of E --> E be discretely approximated by the sequence ((A(n), D(A(n))))(nepsilonN) of operators A(n) where each A(n) is densely defined in the Banach space F-n Let sigma(a)(A) be the approximate point spectrum of A and let sigma(epsilon)(A(n)) denote the epsilon-pseudospectrum of A(n) Generalizing our own result, we show that sigma(a)(A) subset of lim inf sigma(epsilon) (A(n)) = boolean ORnepsilonN boolean AND(kgreater than or equal ton) sigma(epsilon)(A(k)) holds for every epsilon > 0. We deduce that then for every compact set K subset of C lim, dist(sigma(a)(A) boolean AND K, sigma(alpha)(A(n))) = 0 provided there exists M > 0 such that parallel to(lambda - A(n))(-1)parallel to less than or equal to M dist(lambda, sigma(A(n)))(-1) holds for every n and every lambda in the resolvent set rho(A(n)) of A(n). We finally treat the problem under which conditions sigma(a)(A) can be approximated from below. More precisely we investigate the problem: Under which assumptions does boolean AND(epsilon>0) boolean AND(nepsilonN) boolean ORkgreater than or equal ton sigma(epsilon, a)(A(k)) subset of sigma(a)(A) hold where sigma(epsilon,a)(A) denotes the epsilon-approximate pseudospectrum? (C) 2001 Academic Press.

[发布日期] 2001-12-01

[关键词] epsilon-pseudospectrum;discrete convergence;approximation of spectra