Performing certain manipulations on pseudo-differential equations I have come across the following first order equation: $$ D_{t}u-\lambda(t,x,D_{t},D_{x})u=0, \ \ (*) $$ where $\lambda$ is a scalar pseudo-differential operator with the principal symbol being real-valued and independent of $D_{t}.$ But, the lower-order terms of $\lambda$ depend on $D_{t}$ (or $\tau$ at the symbol level).

I was expecting a hyperbolic equation. But, I find that standard text books(like M.Taylor's 'Pseudo-differential operators') treat only equations in which $\lambda$ term is independent of $D_{t}$ (or $\tau$ at the symbol level).

For the equation $(*)$ to be hyperbolic, is it necessary that $\lambda$ to be independent of $D_{t}$? Are there any references which discuss these issues?