On invariant graph subspaces

TL;DR

This paper investigates the decomposition of unbounded 2x2 operator matrices into invariant graph subspaces, providing conditions for such decompositions and introducing a new block diagonalization method.

Contribution

It establishes necessary and sufficient conditions for decomposing operator matrices via invariant graph subspaces and proposes a novel block diagonalization procedure.

Findings

Decomposition occurs if the domain is invariant for angular operators.

A new block diagonalization method is introduced.

Block triangular forms are obtained with a single invariant graph subspace.

Abstract

In this paper we discuss the problem of decomposition for unbounded $2\times 2$ operator matrices by a pair of complementary invariant graph subspaces. Under mild additional assumptions, we show that such a pair of subspaces decomposes the operator matrix if and only if its domain is invariant for the angular operators associated with the graphs. As a byproduct of our considerations, we suggest a new block diagonalization procedure that resolves related domain issues. In the case when only a single invariant graph subspace is available, we obtain block triangular representations for the operator matrices.