Completeness of quasi-normed operator ideals generated by -numbers.

Abstract: We positively resolve Problem 8.2 stated in [A. Pietsch, Traces on operator ideals and related linear forms on sequence ideals (Part I), Indag. Math. (N.S.) (2013) http://dx.doi.org/10.1016/j.indag.2012.08.008]. The question was whether, in the Hilbert space setting, completeness carries over from quasi-normed shift-monotone sequence ideals to the associated quasi-normed operator ideals. In fact, our technique provides even a solution of Problem 14.1.7 in Pietsch’s book “Operator Ideals” (1978). It turns out that all quasi-normed operator ideals over Banach spaces generated by complete quasi-normed symmetric sequence ideals via arbitrary additive -numbers are complete. So the completeness problem is solved completely. [Copyright &y& Elsevier]