Weyl’s theorem for direct sums. (English) Zbl 1164.47019

Let \(T\) be a linear bounded operator on a Hilbert space. The Weyl spectrum of \(A\) is the set \(\sigma_{w}(T)\) of all scalars \(\lambda\) such that \(\lambda I-T\) is not a Fredholm operator of index zero. This set is a subset of the spectrum \(\sigma(T)\) of \(T\). An operator \(T\) satisfies Weyl’s theorem if the complement of \(\sigma_{w}(T)\) in \(\sigma(T)\) coincides with the set of all isolated eigenvalues of \(T\) of finite multiplicity. Let \(S\) and \(T\) be Hilbert space operators. In general, it does not not follow that Weyl’s theorem holds for \(S\oplus T\) if it holds for \(S\) and \(T\). It is proved that, if \(T\) has no isolated points in its spectrum, \(S\) satisfies Weyl’s theorem, and \(\sigma_{w}(T\oplus S) = \sigma(T)\cup\sigma_{w}(S)\), then \(S\oplus T\) satisfies Weyl’s theorem. There are given some sufficient conditions on \(T\) and \(S\) to ensure that such spectral identity holds.


47A53 (Semi-) Fredholm operators; index theories
47B20 Subnormal operators, hyponormal operators, etc.
47A10 Spectrum, resolvent
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