Application of Egorov's Theorem for Pseudodifferential Operators

Question

Let $\;P_{0} \in OPS^{m}_{1,0}(\mathbb{R}^{n} \; \times \; \mathbb{R}^{n})\;$, $\;A \in OPS^{1}(\mathbb{R}^{n} \; \times \; \mathbb{R}^{n})$ and $S(t)$ the solution operator of the scalar hyperbolic equation $$ \frac{\partial u}{\partial t} = i\;A(t,x,Dx)\;u $$ Egorov's theorem applies to the operator $P(t) = S(t)\;P_{0}\;S(t)^{-1}$ (see Theorem 6.1 M. Taylor's notes).

What about $P(t) = S^{-1}(t)\;P_{0}\;S(t)$? Are the principal symbols of $P$ and $A$ related by an Hamiltonian flow? (I couldn't figure out the answer from the proof in the notes)

Answer 1

If $S(t)$ is such that $S(0)=Id$ and $$ \dot S=iA S, $$ the operator $S$ is a Fourier integral operator which quantizes the canonical transformation $\chi$ given by the (non-autonomous) flow of $H_a$, the Hamiltonian vector field of the principal symbol $a$ of $A$, which is assumed of real-principal type. Now if $P_0$ is a pseudodifferential operator of order $m$ with principal symbol $p$, Egorov' theorem says indeed that $S^*P_0S$ is also a pseudodifferential operator of order $m$ with principal symbol $p\circ \chi$. Note in particular that, for $P_0=Id$, you get that the principal symbol of $S^*S$ is 1. Now looking at the operator $SP_0S^*$, you find that it is a pseudodifferential operator with principal symbol $p\circ \chi^{-1}$.