×

Local spectrum and generalized spectrum. (English) Zbl 0739.47002

Let \(A\) be a closed operator with domain \(D(A)\) and range \(R(A)\) in a Hilbert space \(H\). The regular set of \(A\) is the set \(\hbox {reg}(A)=\{\lambda\in\mathbb{C}:\) \(A\) has a generalized resolvent, analytic on a neighbourhood of \(\lambda\}\) where a generalized resolvent of \(A\) in a subset \(U\) of \(\mathbb{C}\) is a bounded operator \(\hbox{Rg}(A,\lambda)\) from \(H\) into \(D(A)\) which is both an inner and outer generalized inverse of \(A-\lambda I\) for \(\lambda\in U\). The set \(\hbox{reg}(A)\) is an open set containing the resolvent set of \(A\) and has analogous properties. This paper contains proofs of results previously announced in [M. Mbekhta, C. R. Acad.Sci., Paris Sér. I 306, No. 14, 593-596 (1988; Zbl 0644.47004)] and also provides some characterizations and properties of Cowen-Douglas operators [M. J. Cowen and R. G. Douglas, Acta Math. 141, 187-261 (1978; Zbl 0427.47016)]. For example, if \(\hbox{Co}(A)\) denotes the largest subspace \(M\) of \(H\) such that \(A(M)=M\), then \(T\in B(H)\) is a Cowen-Douglas operator if and only if \(\hbox{Co}(T)=H\) (alternatively, \(R(T^*)\) is closed and \(N((T^*)^ n)\subset R(T^*)\) for every \(n\in\mathbb{N}\)), \(\hbox{Co}(T^*)=\{0\}\) and \(\dim(T)<\infty\).

MSC:

47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
47B40 Spectral operators, decomposable operators, well-bounded operators, etc.
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ion Colojoară and Ciprian Foiaş, Theory of generalized spectral operators, Gordon and Breach, Science Publishers, New York-London-Paris, 1968. Mathematics and its Applications, Vol. 9. · Zbl 0189.44201
[2] M. J. Cowen and R. G. Douglas, Complex geometry and operator theory, Acta Math. 141 (1978), no. 3-4, 187 – 261. · Zbl 0427.47016 · doi:10.1007/BF02545748
[3] Jean-Philippe Labrousse, Les opérateurs quasi Fredholm: une généralisation des opérateurs semi Fredholm, Rend. Circ. Mat. Palermo (2) 29 (1980), no. 2, 161 – 258 (French, with English summary). · Zbl 0474.47008 · doi:10.1007/BF02849344
[4] Mostafa Mbekhta, Généralisation de la décomposition de Kato aux opérateurs paranormaux et spectraux, Glasgow Math. J. 29 (1987), no. 2, 159 – 175 (French). · Zbl 0657.47038 · doi:10.1017/S0017089500006807
[5] -, Sur la théorie spectrale généralisée, C. R. Acad. Sci. Paris 306 (1988), 593-596. · Zbl 0644.47004
[6] Mostafa Mbekhta, Résolvant généralisé et théorie spectrale, J. Operator Theory 21 (1989), no. 1, 69 – 105 (French). · Zbl 0694.47002
[7] -, Théorie spectrale locale et limite de nilpotents, Proc. Amer. Math. Soc. (to appear).
[8] Florian-Horia Vasilescu, Analytic functional calculus and spectral decompositions, Mathematics and its Applications (East European Series), vol. 1, D. Reidel Publishing Co., Dordrecht; Editura Academiei Republicii Socialiste România, Bucharest, 1982. Translated from the Romanian. · Zbl 0495.47013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.