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Sur la théorie spectrale locale et limite des nilpotents. (On local spectral theory and limit of nilpotents). (French) Zbl 0721.47006

For \(\lambda_ 0\in {\mathbb{C}}\) a closed linear operator A in a Banach space X with domain D(A) is said to have the single-valued extension property at \(\lambda_ 0\) if \(f(\lambda)=0\) is the only solution of the equation \((\lambda I-A)f(\lambda)=0\) that is analytic in a neighborhood of \(\lambda =\lambda_ 0\). The paper under review gives a necessary and sufficient condition for A to have this property, introducing a subspace (“le coeur analytique”) K(A) of X for A:
\(K(A)=\{u\in X;\) there exist \(a>0\) and \(\{v_ n\}_{n\geq 0}\subset D(A)\) such that \(v_ 0=0\), \(Av_{n+1}=v_ n\) and \(\| v_ n\| \leq a^ n\| u\|\) for all \(n\geq 0\}.\)
Various spectral and related properties of bounded linear operators T on a Hilbert space with \(K(T)=\{0\}\) are also studied. Among others it is shown that 0 belongs to the spectrum \(\sigma (T_{| M})\) of the restriction \(T_{| M}\) of T to each invariant subspace \(M\neq \{0\}\), and that the set \({\mathcal K}{\mathcal A}\) of the bounded operators T with \(K(T)=K(T^*)=\{0\}\) is a proper subset of the closure (\(\tilde {\mathcal N})\) of the nilpotent operators, while each S in (\(\tilde {\mathcal N})\) is expressed as \(S=T_{\epsilon}+K_{\epsilon}\) for every \(\epsilon >0\) and some \(T_{\epsilon}\) in \({\mathcal K}{\mathcal A}\) and some compact operator \(K_{\epsilon}\) with \(\| K_{\epsilon}\| <\epsilon\).

MSC:

47A10 Spectrum, resolvent
47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
47B40 Spectral operators, decomposable operators, well-bounded operators, etc.
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