Joint spectra of strongly hyponormal operators on Banach spaces (Q749846)

Let \(T=H+iK\) be a bounded linear operator on a complex Banach space X decomposed by two hermitian operators H and K (i.e. their spatial numerical range is real). then \(\bar T:=H-iK\), T is called hyponormal iff the spatial numerical range of i(HK-KH) is contained in the nonnegative reals, and iff additionally \(H^ 2\) and \(K^ 2\) are hermitian T is called strongly hyponormal. Two such operators are called doubly commuting iff \(T_ 1T_ 2=T_ 2T_ 1\) and \(\bar T_ 1T_ 2=T_ 2\bar T_ 1\). For a doubly commuting n-tuple \({\mathcal T}=(T_ 1,...,T_ n)\) of hyponormal operators let \(\sigma\) (\({\mathcal T})\) denote the Taylor spectrum (i.e. the set of \(z=(z_ 1,...,z_ n)\in {\mathbb{C}}^ n\) such that the Koszul complex associated to \({\mathcal T}-z\) is not exact), \(\sigma_{\pi}({\mathcal T})\) the joint approximate point spectrum, \(\sigma_{cs}({\mathcal T})\) the complete star spectrum (i.e. the set of \(z\in {\mathbb{C}}^ n\) such that \[ \sum^{k}_{\mu =1}\overline{(T_{j_{\mu}}-z_{j_{\mu}})}(T_{j_{\mu}}- z_{j_{\mu}})+\sum^{m}_{\nu =1}(T_{\ell_{\nu}}- z_{\ell_{\nu}})\overline{(T_{\ell_{\nu}}-z_{\ell_{\nu}})} \] is not invertible for some partition \(\{j_ 1,...,j_ k\}\cup \{\ell_ 1,...,\ell_ m\}\) of \(\{1,...,n\}\)), and \(\sigma_ r({\mathcal T})\) the right spectrum (i.e. the set of \(z\in {\mathbb{C}}^ n\) such that \(\sum^{n}_{j=1} (T_ j-z_ j)\overline{(T_ j-z_ j)}\) is not invertible). If in this situation x is uniformly convex, \(\sigma\) (\({\mathcal T})\) is shown to be contained in \(\sigma_{cs}({\mathcal T})\). If additionally \(T_ 1,...,T_ n\) are strongly hyponormal, then \(\sigma_{cs}({\mathcal T})= \sigma_ r({\mathcal T})= \{z\in {\mathbb{C}}^ n:\bar z\in \sigma_{\pi}(\bar{\mathcal T})\}\), and if moreover X is uniformly smooth, then \(\sigma({\mathcal T})\) equals \(\sigma_{cs}({\mathcal T})\).