Exercise 33, § 3

Let $m\geqslant 1$ be an integer.

a) There exists a unique linear mapping $A^+$ (resp. $A^-$) of $H^m(\mathbf{R})$ into $H^{m-1}(\mathbf{R})$ such that

$A^+f(x) =f'(x)-xf(x)$ (resp. $A^-f(x) =f'(x) +xf(x)$ )

for every $f\in \mathscr{D}(\mathbf{R})$ and every $x\in \mathbf{R}$.

b) The mapping $A^+$ is a Fredholm mapping of index 1 and $A^-$ is a Fredholm mapping of index $-1$. (Consider the image of the Hermite functions under $A^+$ and $A^-$, cf. exercise 3 of II, p. 263.)