Decay spectrum and decay subspace of normal operators

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6 Citations (Scopus)

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 A(p), A(ac) and A(sc) such that there exists an orthonormal basis of eigenvectors for the operator A(p) the operator A(ac) has purely absolutely continuous spectrum and the operator A(sc) has purely singular continuous spectrum. We show the existence of a natural further decomposition of the singular continuous component A c into a direct sum of two self-adjoint operators A(sc)(D) and A(sc)(ND). 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.
Original languageEnglish
Pages (from-to)1245-1255
Number of pages11
JournalProceedings of the Royal Society of Edinburgh. Section A. Mathematics
Volume131
Issue number4
Publication statusPublished - 2001

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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