Graph Subspaces and the Spectral Shift Function

Abstract

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.

The Singularly Continuous Spectrum and Non-Closed Invariant Subspaces

“…It is not hard to see (cf., e.g., Example 1 in [2]) that the set S pp coincides with the set of all eigenvalues of the operator B. Hence, by Theorem 3.3 one proves that S pp coincides with the set of all atoms of the measure ω.…”

Bounds on Variation of Spectral Subspaces under J-Self-adjoint Perturbations

“…Surely, one can also associate with the matrix L another operator Riccati equation, 11) assuming that a solution K ′ (if it exists) should be a bounded operator from H 1 to H 0 . It is well known that the solutions to the Riccati equations (2.1) and (2.11) determine invariant subspaces for the operator matrix L (see, e.g., [5] for the case where the matrix L is self-adjoint or [31] for the case of a non-self-adjoint L). These subspaces have the form of the graphs…”

“…We start by recalling the concepts of weak, strong, and operator solutions to the operator Riccati equation (see [5,6] A bounded operator K ∈ B(H 0 , H 1 ) is said to be a weak solution of the Riccati equation…”

“…Recall that if M and N are subspaces of a Hilbert space, the operator angle Θ(M, N) between M and N measured relative to the subspace M is introduced by the following formula [26]: 5) where I M denotes the identity operator on M and P M and P N stand for the orthogonal projections onto M and N, respectively. 6) and assume that V < δ /2.…”

“…Fixed-point based approaches to prove the solvability of the operator Riccati equation with bounded entries A 0 and A 1 have been used in many papers (see, e.g., [2], [18], [42], [46], [47]). In the case where at least one of the entries A 0 and A 1 is an unbounded self-adjoint or normal operator, a fixed-point approach has been employed in [5], [6], [36], and [40]. …”