Tag Archives: stupidity

Alok’s question

So, in class today, we were talking about what it means for a (constant coefficient) operator to be elliptic.   If \(D\) is such an operator, of order \(m\), say

\[ D = \sum_{|\alpha|\le m} L_\alpha \frac{\partial^\alpha}{\partial x^\alpha} , \]

then the correct definition of ellipticity is that the principal symbol

\[ \sigma_D^m = \sum_{\alpha = m} L_\alpha \xi^\alpha \]

should be invertible for all \(\xi\neq 0\).   This of course implies that the total symbol (which we just called the symbol)

\[ \sigma_D = \sum_{\alpha \le m} L_\alpha \xi^\alpha \]

is invertible for \(\xi\) sufficiently large, and in class I incautiously stated that the reverse implication is true as well.  Alok correctly questioned this, pointing out that lower order terms could tip a non-invertible principal symbol over into invertibility: a simple explicit example is the symbol

\[ \left(\begin{array}{cc} 1 & 0\\ 0 & 0\end{array}\right) \xi^2 +  \left(\begin{array}{cc} 0 & 0\\ 0 & 1\end{array}\right) \xi\]
which should not be classed as elliptic.  (In my defense, I will say that there is a sense in which it is true that “an operator is elliptic if and only if its total symbol is invertible for large \(\xi\)” – as we will see when we discuss pseudodifferential operators – but the word “invertible” has to be interpreted in a more sophisticated way than mere pointwise invertibility for each \(\xi\) individually.)   Apologies for the confusion here, and thanks to Alok for keeping me on track.