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Spectra of selfadjoint operators in constructive analysis. (English) Zbl 0852.47010

Summary: The approximate point spectrum \(\sigma_a(T)\) of a selfadjoint operator \(T\) on a nontrivial separable Hilbert space is examined constructively with the help of the functional calculus for \(T\). In particular, it is proved that \(\sigma_a(T)\) is compact if and only if \(|f(T)|\) can be computed for each \(f\) in \({\mathcal C}[- b, b]\), where \(b> 0\) is a bound for \(T\).
The paper culminates in a full constructive analysis of the spectrum of a compact selfadjoint operator with infinite-dimensional range. Brouwerian examples show that the results are the best possible in the constructive setting.

MSC:

47B25 Linear symmetric and selfadjoint operators (unbounded)
47A10 Spectrum, resolvent
47A60 Functional calculus for linear operators
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References:

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