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I am currently reading about pseudodifferential operators and their symbols, and I came across the notion of classical pseudodifferential operators. For these it is possible to find an asymptotic expansion of the symbol in terms of positively homogeneous functions.

So, according to the notes that I am reading we say that a pseudodifferential operator $A \in \Psi^m$ is classical if the symbol $\sigma_A(x, \xi)$ admits an asymptotic expansion

\begin{equation} \sigma_A(x,\xi) \backsim \sum_{k \geq 0} a_{m - k} \, (x,\xi) \qquad (|\xi| \to \infty) \end{equation}

where each $a_{m - k}$ satisfies the equation \begin{equation} a_{m - k}\,\,(x,\lambda \xi) = \lambda^{(m - k)} \,\, a_{m - k}\,\,(x,\xi)\quad \text{for } \lambda > 0 \end{equation}

Now, there is a comment in the notes that such an expansion has the special property of being unique (as opposed to general asymptotic expansions where the terms are not positively homogeneous). Unfortunately I cannot deduce myself why this is so. If someone could give some hints or details of the reason that would be really helpful, thanks a lot!

EDIT: One guess I have is that, instead of the uniqueness of the expansion, what is acutally unique (or well defined in the context of positively homogeneous terms) is the leading homogeneous term $a_m$, i.e. the principal symbol. This is not the case of a general asymptotic expansion, that's why I suppose that this is what the lecture notes are referring to .. might that be a correct interpretation of the comment I mention ?

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