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Local spectral theory of shifts with operator-valued weights. (Sur la théorie spectrale locale des shifts à poids opérateurs.) (French) Zbl 1100.47028
Let $$H$$ be an infinite-dimensional separable complex Hilbert space, let $$L(H)$$ be the algebra of bounded linear operators over $$H$$, and let $$H^{(\infty)}$$ be the Hilbert space direct sum, that is, $$H^{(\infty)}:=\{(x_n)_{n\geq0}:\forall n\geq0,\;x_n\in H$$ and $$\sum_{n\geq0}\| x_n\| ^2<\infty\}$$. An operator $$T\in L(H)$$ is said to be a weighted shift with weights $$(T_n)_{n\geq0}$$ if for all $$x=(x_n)_{n\geq0}$$ of $$H^{(\infty)}$$, $$Tx=(0,T_0x_0,T_1x_1,\dots,T_nx_n,\dots)$$ and, in addition, $$\sup_{n\geq0}\| T_n\| <\infty$$.
In the paper under review, some local spectral properties of these operators are studied. The authors state that the adjoint of a weighted shift $$T$$ with invertible weights $$(T_n)_{n\geq0}$$ enjoys the single-valued extension property (SVEP) if and only if $$R_2(T):=\sup\{| \lambda| :\;\lambda\in\sigma_p(T^*)\}=0$$. In addition, it is proved that if $$r_2(T)=r(T)$$, $$T$$ enjoys the Dunford property (C), where $$r_2(T):=1/\limsup_{n\to\infty}\| \pi(n)^{-1}\| ^{1/n}$$ and $$\pi(n):=T_{n-1}T_{n-2}\cdots T_0$$. Moreover, if $$R_1(T):=\liminf_{n\to\infty}\left(\inf_{k\geq0}\| \pi(n+k)\pi(k)^{-1}\| \right)^{1/n}<r_2(T)$$, $$T$$ does not satisfy the Bishop property ($$\beta$$). Finally, it is established that a weighted shift $$T$$ with not necessarily invertible weights is decomposable if and only if it is quasinilpotent.

##### MSC:
 47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.) 47A11 Local spectral properties of linear operators 47B40 Spectral operators, decomposable operators, well-bounded operators, etc.
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