Graph Subspaces and the Spectral Shift Function

[摘要] We obtain a new representation for the solution to the operatorSylvester equation in the form of a Stieltjes operator integral.We also formulate new sufficient conditions for the strongsolvability of the operator Riccati equation that ensures theexistence of reducing graph subspaces for block operator matrices.Next, we extend the concept of the Lifshits-Krein spectral shiftfunction associated with a pair of self-adjoint operators to thecase of pairs of admissible operators that are similar toself-adjoint operators. Based on this new concept we express thespectral shift function arising in a perturbation problem for blockoperator matrices in terms of the angular operators associated withthe corresponding perturbed and unperturbed eigenspaces.

[学科分类] 数学（综合）