2013/07/24 by Konstantin A. Makarov, Makarov, Konstantin A., Stephan Schmitz +3 · 1 citation
Mathematics · #47A15 #47A55 #47A62 #FOS: Mathematics #Functional Analysis (math.FA) #Spectral Theory (math.SP) #math.FA #math.SP #msc:47A15 #msc:47A55 #msc:47A62
paper · pdf · doi:10.48550/arxiv.1307.6439
17 pages
arxiv created 2013/07/24 · arxiv updated 2013/10/18
The problem of block diagonalization for diagonally dominant symmetric block operator matrices with self-adjoint diagonal entries is considered. We show that a reasonable block diagonalization with respect to a reducing graph subspace requires a related skew-symmetric operator to be a strong solution to the associated Riccati equation. Under mild additional regularity conditions, we also establish that this skew-symmetric operator is a strong solution to the Riccati equation if and only if the graph subspace is reducing for the given operator matrix. These regularity conditions are shown to be automatically fulfilled whenever the corresponding relative bound of the off-diagonal part is sufficiently small. This extends the results by Albeverio, Makarov, and Motovilov in [Canad. J. Math. Vol. 55, 2003, 449--503], where the off-diagonal part is required to be bounded.