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Decay spectrum and decay subspace of normal operators

Published online by Cambridge University Press:  12 July 2007

I. Antoniou  and
S. A. Shkarin
I. Antoniou
Affiliation:
International Solvay Institutes for Physics and Chemistry, Campus Plaine ULB, CP 231, Bd. du Triomphe, Brussels 1050, Belgium
S. A. Shkarin
Affiliation:
Moscow State University, Department of Mathematics and Mechanics, 119899 Moscow, Russia, Vorobjovy Gory
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Abstract

Let A be a self-adjoint operator on a Hilbert space. It is well known that A admits a unique decomposition into a direct sum of three self-adjoint operators Ap, Aac and Asc such that there exists an orthonormal basis of eigenvectors for the operator Ap, the operator Aac has purely absolutely continuous spectrum and the operator Asc has purely singular continuous spectrum. We show the existence of a natural further decomposition of the singular continuous component Asc into a direct sum of two self-adjoint operators and . The corresponding subspaces and spectra are called decaying and purely non-decaying singular subspaces and spectra. Similar decompositions are also shown for unitary operators and for general normal operators.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 131 , Issue 6 , December 2001 , pp. 1245 - 1255
Copyright
Copyright © Royal Society of Edinburgh 2001

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