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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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