Inertia theorems for operator Lyapunov inequalities
Section snippets
Introduction and the main result
The inertia of a square matrix is the triple (ν(A),ζ(A),π(A)), wherewhere and . Inertia theorems for matrices concern relations between the inertia of Hermitian solutions Q of the Lyapunov equationand the matrix A. The fundamental result was by Ostrowski and Schneider [12], and later contributions can be found in [15], [2]. We shall
Preliminaries on indefinite inner products and proof of the main result
The proof of our main theorem relies on the fact that any self-adjoint solution of the operator Lyapunov inequality gives rise to a natural indefinite inner product space. So, we will first state a few preliminaries and results about indefinite inner product spaces which will be used in the proof. For more details, see [1].
Let be a vector space over . An indefinite inner product [·,·] on is a map satisfying:
- 1.
[αx1+βx2,y]=α[x1,y]+β[x2,y], and .
- 2.
for all
Corollaries
In this section, we give a few corollaries of our main theorem applied to Lyapunov equations with possibly unbounded observation operators.
Throughout this section, we assume that X is a Hilbert space and is the infinitesimal generator of a C0-semigroup {T(t)}t⩾0 on X. Definition 3.1 Let us denoteA satisfies the spectrum decomposition assumption if σ+(A) is a bounded set which is separated from σ−(A) in such a way that a rectifiable, simple closed curve, Γ, can be drawn so
Acknowledgements
The authors thank the referee for his careful reading of the manuscript and, in particular, for pointing out a gap in the original manuscript.
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