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Graph Subspaces and the Spectral Shift Function

  Published:2003-06-01
 Printed: Jun 2003
  • Sergio Albeverio
  • Konstantin A. Makarov
  • Alexander K. Motovilov
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Abstract

We obtain a new representation for the solution to the operator Sylvester equation in the form of a Stieltjes operator integral. We also formulate new sufficient conditions for the strong solvability of the operator Riccati equation that ensures the existence of reducing graph subspaces for block operator matrices. Next, we extend the concept of the Lifshits-Krein spectral shift function associated with a pair of self-adjoint operators to the case of pairs of admissible operators that are similar to self-adjoint operators. Based on this new concept we express the spectral shift function arising in a perturbation problem for block operator matrices in terms of the angular operators associated with the corresponding perturbed and unperturbed eigenspaces.
MSC Classifications: 47B44, 47A10, 47A20, 47A40 show english descriptions Accretive operators, dissipative operators, etc.
Spectrum, resolvent
Dilations, extensions, compressions
Scattering theory [See also 34L25, 35P25, 37K15, 58J50, 81Uxx]
47B44 - Accretive operators, dissipative operators, etc.
47A10 - Spectrum, resolvent
47A20 - Dilations, extensions, compressions
47A40 - Scattering theory [See also 34L25, 35P25, 37K15, 58J50, 81Uxx]
 

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