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On the weak decomposition property \((\delta_w)\). (English) Zbl 1202.47005
Let \(T\) be a bounded linear operator on a complex Banach space \(X\). For a closed subset \(F\) of \(\mathbb{C}\), let \(\mathcal{X}_T(F)\) be the corresponding glocal spectral subspace, the set of all \(x\in X\) for which there is an analytic function \(f:\mathbb C\backslash F\to X\) such that \((T-\lambda)f(\lambda)=x\) for all \(\lambda\in\mathbb C\backslash F\). The operator \(T\) is said to have weak spectral decomposition property, weak-SDP for short, if for every finite open covering \(\{U_1,\dots ,U_n\}\) of \(\mathbb{C}\), there exist closed \(T\)-invariant subspaces \(X_1,\dots ,X_n\) such that
\[ \sigma(T_{| X_i})\subset U_i,~~(1\leq i\leq n ),~~\overline{X_1+\dots +X_n}=X, \] where \(\sigma(\cdot)\) denotes, as usual, the spectrum. It is said to have decomposition property \((\delta)\) if
\[ \mathcal{X}_T(\overline{U})+\mathcal{X}_T(\overline{V})=X \] for every open covering \(\{U,V\}\) of \(\mathbb{C}\). In [Proc. London Math. Soc. (3) 75, 323–348 (1997; Zbl 0881.47007)], E. Albrecht and J. Eschmeier proved that \(T\) enjoys property \((\delta)\) exactly when it is the quotient of a decomposable operator by one of its closed invariant subspaces, and that \(T\) is decomposable if and only if both \(T\) and \(T^*\) have this property (see also the monograph by K. B. Laursen and M. M. Neumann [“An introduction to local spectral theory” (London Mathematical Society Monographs, New Series 20; Oxford: Clarendon Press) (2000; Zbl 0957.47004)]).
In the present paper, the authors introduce and study local spectral properties of operators with what they call weak decomposition property \((\delta_w)\). An operator \(T\in\mathcal{L}(X)\) is said to have such property at a point \(\lambda\in\mathbb C\) if there is \(r(\lambda)>0\) such that for every \(0\leq r\leq r(\lambda)\) and for every finite open covering \(\{U_1,\dots ,U_n\}\) of \(\mathbb C\) with \(\sigma(T)\backslash\{\mu\in\mathbb{C}:| \mu-\lambda| <r\}\subset U_1 \), the sum \[ \mathcal{X}_T(\overline{U_1})+\dots +\mathcal{X}_T(\overline{U_n}) \] is dense in \(X\). It is said to have the weak decomposition property \((\delta_w)\) provided that this property holds for \(T\) at every point \(\lambda\in\mathbb{C}\). From the above definitions, it follows that, if an operator enjoys weak-SDP or property \((\delta)\), then it possesses property \((\delta_w)\). The authors give, in particular, an example of a unilateral weighted shift operator with infinitely many zero weights possessing property \((\delta_w)\) but without weak-SDP, property \((\delta)\) and Dunford’s condition \((C)\). They also prove that, if an operator \(T\in\mathcal{L}(X)\) has property \((\delta_w)\) at a point \(\lambda\in\mathbb{C}\), then \(T^*\) enjoys the single-valued extension property at \(\lambda\). Furthermore, they give some elementary results regarding localizable spectrum and support points set of operators with property \((\delta_w)\), and obtain some applications to multipliers on semi-simple commutative Banach algebras with property \((\delta_w)\).
The reviewer would like to point out that a unilateral weighted shift with infinitely many zero weights has always property \((\delta_w)\), and enjoys weak-SDP, or property \((\delta)\), or Dunford’s condition \((C)\) precisely when it is quasi-nilpotent; see, for example, the paper by P. Aiena and M. T. Biondi [Mat. Vesn. 54, No. 3–4, 57–70 (2002; Zbl 1079.47001)] and the one by the reviewer [Stud. Math. 163, 41–69 (2004; Zbl 1070.47023)].

MSC:
47A10 Spectrum, resolvent
47A11 Local spectral properties of linear operators
47B40 Spectral operators, decomposable operators, well-bounded operators, etc.
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