Schur complement dominant operator matrices

Abstract

We propose a method for the spectral analysis of unbounded operator matrices in a general setting which fully abstains from standard perturbative arguments. Rather than requiring the matrix to act in a Hilbert space H, we extend its action to a suitable distributional triple D ⊂ H ⊂ D − and restrict it to its maximal domain in H. The crucial point in our approach is the choice of the spaces D and D − which are essentially determined by the Schur complement of the matrix. We show spectral equivalence between the resulting operator matrix in H and its Schur complement, which allows to pass from a suitable representation of the Schur complement (e.g. by generalised form methods) to a representation of the operator matrix. We thereby generalise classical spectral equivalence results imposing standard dominance patterns.The abstract results are applied to damped wave equations with possibly unbounded and/or singular damping, to Dirac operators with Coulomb-type potentials, as well as to generic second order matrix differential operators. By means of our methods, previous regularity assumptions can be weakened substantially.

Resolvent estimates for the one-dimensional damped wave equation with unbounded damping

“…In order to properly define such a matrix as an unbounded (non-self-adjoint) operator, we follow Section 4 of [18] which specialises a new general framework for the spectral analysis of operator matrices to the particular case of the DWE. Assuming that a, q ∈ L 1 loc (R) with a, q ≥ 0 a.e., let…”

“…and let D * S be the space of bounded, conjugate-linear functionals on D S . It can be shown that the canonical embeddings D S ⊂ H 2 ⊂ D * S are continuous with dense range, that C ∞ c (R) is densely contained in D S and that D S can also be continuously embedded in H 1 (see [18,Prop. 4.6]).…”