Graph Subspaces and the Spectral Shift Function
Printed: Jun 2003
Konstantin A. Makarov
Alexander K. Motovilov
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.
47B44 - Accretive operators, dissipative operators, etc.
47A10 - Spectrum, resolvent
47A20 - Dilations, extensions, compressions
47A40 - Scattering theory [See also 34L25, 35P25, 37K15, 58J50, 81Uxx]