Reducing graph subspaces and strong solutions to operator Riccati equations

TL;DR

This paper investigates block diagonalization of symmetric block operator matrices, establishing conditions under which a skew-symmetric operator solves the Riccati equation, extending previous bounded off-diagonal results to more general cases.

Contribution

It extends existing results by showing that regularity conditions for Riccati solutions hold automatically under small off-diagonal bounds, broadening the applicability.

Findings

A skew-symmetric operator is a strong Riccati solution if and only if the graph subspace reduces the operator.

Regularity conditions are automatically satisfied when the off-diagonal part has sufficiently small relative bound.

The results generalize previous bounded off-diagonal cases to unbounded scenarios.

Abstract

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. \textbf{55}, 2003, 449--503], where the off-diagonal part is required to be bounded.