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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.

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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