# Estimates for Non-Selfadjoint Operators

Chapter

## Abstract

We have seen in the first chapter that it is very easy to prove a priori estimates for pseudo-differential operators of real principal type (see, e.g., the proof of Theorem 1.2.38), and also for principal-type operators whose imaginary part is either always non-negative or always non-positive (Theorem 1.2.39). Thanks to Remark 1.2.36, it boils down to proving an depending continuously on choose λ large enough so that, thanks to Gå;rding’s inequality (Theorem 1.1.26),

*L*^{2}−*L*^{2}injectivity estimate |*Lu*|_{0}∼|*u*|_{0}for$$
L = D_t + a(t,x,\xi )^w + ib(t,x,\xi )^w , where a,b \in S_{1,0}^1 , b \geqslant 0 (or b \leqslant 0)
$$

(3.1.1)

*t*∈ℝ. Note that Lemma 4.3.21 in the appendix provides an even more precise form of such an estimate.1 At any rate, proving local solvability or an*L*^{2}−*L*^{2}injectivity estimate for that class of examples does not require much: for instance in the case*b*≥0, just conjugate the operator*L*by*e*^{λ}*t*so that$$
L_\lambda = e^{2\pi \lambda t} Le^{ - 2\pi \lambda t} = D_t + a(t,x,\xi )^w + i(b(t,x,\xi )^w + \lambda ),
$$

*b*^{ w }+λ≥0 as an operator, and apply Lemma 4.3.21. Note in particular that for an evolution equation (*t*∈ ℝ is the time-variable) where Re*Q*(*t*) is a pseudo-differential operator with real-valued Weyl symbol, the inequality (4.3.47) shows that the forward Cauchy problem is well-posed whenever Re*Q*(*t*) is bounded from below, e.g., is non-negative. Most evolution equations or systems of mathematical physics are essentially of the type (3.1.2) with a non-negative Re*Q*(*t*) together with very significant complications coming from rough coefficients or nonlinearities.## Keywords

Symplectic Manifold Principal Symbol Principal Type Local Solvability Weyl Symbol
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