Separation of two (possibly unbounded) components of the spectrum of a linear operator |
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Authors: | Giovanni Dore Alberto Venni |
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Institution: | (1) Dipartimento di Matematica, Università di Bologna, Piazza di Porta S.Donato 5, 40127 Bologna, Italy |
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Abstract: | LetX be a complex Banach space andA: D(A)X a densely defined closed linear operator whose resolvent set contains the real line and for which (–A)–1 is bounded onR. We give a necessary and sufficient condition, in terms of the complex powers ofA and –A, for the existence of a decompositionX=X
+X
–, whereX
± are closed subspaces, invariant forA, the spectra of the reduced operatorsA
± are {(A);Im>0} and {(A);Im<0} respectively, and (–A
±)–1 is bounded forIm0.Finally we give an example of an operator in anL
p-type space for which the decomposition exists if 1<p<+ and does not exist ifp=1. |
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Keywords: | |
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