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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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