A Lyapunov lemma for elliptic systems (Q1426971)

The authors of this very well written article prove an interesting result within the functional calculus of classical pseudo-differential operators. More precisely, they fix an \(N\times N\) system \(A\) of classical pseudo-differential operators of order \(\leq m\) on a compact smooth manifold and they assume that \(A\) is elliptic. Their aim is to answer the following questions: 1. Under what conditions on \(A\) is there an \(N \times N\) system \(T\) of classical pseudo-differential operators of order 0 on \(X\) so that 1) \(T\) is (formally) self-adjoint and positive, and 2) there exists a constant \(C>0\) such that \(\text{Re} (Af,Tf)_2\geq\| f \|^2_{m/2}\) for every function \(f\in C^\infty (X;\mathbb{C}^N)\)? 2. How can such a system \(T\) be constructed? The authors first identify a very natural necessary condition for \(T\) to exist. This necessary condition asserts that the eigenvalues of the principal symbol matrix \(a_m(x, \xi)\), for each \((x,\xi)\in T^*(X)\setminus 0\), as well as the spectrum (which consists of eigenvalues) of \(A\) as an operator on \(L^2(X,\mathbb{C}^N)\) with domain in the Sobolev space \(H^m(X,\mathbb{C}^N)\), must have positive real part. It turns out that this condition is also sufficient. Indeed, under this condition, the authors obtain \(T\) by stating and proving a Lyapunov lemma applied to the system \(A\). The Lyapunov lemma for matrices states that given an \(N\times N\) complex matrix \(A\) with eigenvalues having positive real part, the map \(T\to TA+A^*T\) is an isomorphism on the space of \(N\times N\) complex matrices with an explicit inverse. Moreover, given a Hermitian positive matrix \(Q\) there exists a Hermitian positive matrix \(T\) such that \(Q=TA+A^*T\). The proof of the infinite dimensional result mimics the proof of the finite dimensional case, with several technical complications that the authors resolve elegantly. As the authors observe, one particular interest of their construction of \(T\) is that \(T\) is obtained as a classical pseudo-differential operator. That is to say, the symbol is written as a sum of homogeneous terms, and thus, it is completely determined.