On spectra of some linear elliptic differential operators (Q2487456)

Let \(X\) be a Hilbert space with the norm \(\| \cdot\| \), \(\mathcal B(X)\) the Banach algebra of bounded linear operators \(A:X\to X\), \(L^2\) the Lebesgue space of strongly measurable (in the sense of Bochner) functions \(u: \mathbb R^n\to X\) with the usual inner product \((\cdot,\cdot)_0\), and \(H^m= \{u\in L^2 : D^\alpha u\in L^2 \text{ for } | \alpha | \leq m\}\) (\(m\in \mathbb Z_+\)) the Sobolev space equipped with the standard structure of Hilbert space. Consider a linear differential operator \(P=\sum_{| \alpha | \leq m}A_\alpha (x)D^\alpha \), where \(m\) is even, the leading coefficients \(A_\alpha\), \(| \alpha | =m\), are constant operators, and for \(| \alpha | <m\) the coefficients \(A_\alpha\) are continuous, bounded functions \(u: \mathbb R^n\to \mathcal B(X)\) with continuous, bounded derivatives up to \(m\)th order. Denote by \(P_m(\xi)\) the principal symbol of the operator \(P\) (\(\xi\in\mathbb R^n)\); \(P_m^*(\xi )\) is the principal symbol of the dual (formally conjugate) operator \(P_+\) for P, acting by the formula \[ P_+u=\sum_{| \alpha | \leq m}(-1)^{| \alpha | }D^\alpha (A_\alpha^*(x)u(x)), \] where \(u\in H^m\) and \(A_\alpha^*(x)\) is the conjugate operator for \(A_\alpha (x)\). Suppose the operator P is strongly elliptic on \(\mathbb R^n\); this means that the following inequality holds: \[ (-1)^{m/2}((P_m(\xi)+P_m^*(\xi))h,h)_0/2\geq C| \xi | ^m\| h\| ^2\qquad \forall \xi\in\mathbb R^n\qquad\forall h\in X, \] where \(C\) is a positive constant. The main result is that the spectrum of \(P:H^m\to L^2\) lies in a half--plane \(\Re\lambda>\Theta\), \(\Theta\in\mathbb R\).