Spectral approximants of normal operators

For each non-empty subset Λ of the complex plane, let (Λ) be the set of all those operators (on a fixed Hilbert space H) whose spectrum is included in Λ. The problem of spectral approximation is to determine how closely each operator on H can be approximated (in the norm) by operators in (Λ). The pr...

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Bibliographic Details
Published in:Proceedings of the Edinburgh Mathematical Society 1974-03, Vol.19 (1), p.51-58
Main Author: Halmos, P. R.
Format: Article
Language:English
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Summary:For each non-empty subset Λ of the complex plane, let (Λ) be the set of all those operators (on a fixed Hilbert space H) whose spectrum is included in Λ. The problem of spectral approximation is to determine how closely each operator on H can be approximated (in the norm) by operators in (Λ). The problem appears to be connected with the stability theory of certain differential equations. (Consider the case in which Λ is the right half plane.) In its general form the problem is extraordinarily difficult. Thus, for instance, even when Λ is the singleton {0}, so that (Λ) is the set of quasinilpotent operators, the determination of the closure of (Λ) has been an open problem for several years (3, Problem 7).
ISSN:0013-0915
1464-3839
DOI:10.1017/S0013091500015364