2023/10/12 by Lindner, Marko, Schmeckpeper, Dennis
#47-08 #47A10 #FOS: Mathematics #Functional Analysis (math.FA) #Numerical Analysis (math.NA) #Secondary 47A25 #Spectral Theory (math.SP)
paper · doi:10.48550/arxiv.2310.08447
The stability of an approximating sequence (An) for an operator A usually requires, besides invertibility of A, the invertibility of further operators, say B, C, …, that are well-associated to the sequence (An). We study this set, \A,B,C,…\, of so-called stability indicators of (An) and connect it to the asymptotics of ‖An‖, ‖An-1‖ and κ(An)=‖An‖‖An-1‖ as well as to spectral pollution by showing that \limsup \rm Specε An= \rm Specε A∪\rm Specε B∪\rm Specε C∪…. We further specify, for each of ‖An‖, ‖An-1‖, κ(An) and \rm Specε An, under which conditions even convergence applies.