The Taylor joint spectrum and restriction to hyperinvariant subspaces

It is well known that for a single bounded operator \(A_0\) on a Hilbert \(\mathfrak{H}\), if \(\mathfrak{M}\subset \mathfrak{H}\) is hyperinvariant for \(A_0\), then the spectrum of \(A_0|_{\mathfrak{M}}\) is contained in the spectrum of \(A_0\). In this note, we modify an example of Taylor to prov...

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Published in:arXiv.org 2019-11
Main Author: Timko, Edward J
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description It is well known that for a single bounded operator \(A_0\) on a Hilbert \(\mathfrak{H}\), if \(\mathfrak{M}\subset \mathfrak{H}\) is hyperinvariant for \(A_0\), then the spectrum of \(A_0|_{\mathfrak{M}}\) is contained in the spectrum of \(A_0\). In this note, we modify an example of Taylor to prove the following. There exist a quadruple \(A=(A_1,A_2,A_3,A_4)\) of commuting bounded Hilbert space operators and a hyperinvariant subspace \(\mathfrak{X}_1\) for \(A\) such that the Taylor joint spectrum of \(A\) restricted to \(\mathfrak{X}_1\) is a not a subset of the Taylor joint spectrum of \(A\).
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title The Taylor joint spectrum and restriction to hyperinvariant subspaces
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