  We characterize the action of isotropic pseudodifferential operators on
functions in terms of their action on Hermite functions. We show that an
operator $A : S(\mathbb{R}) \to S(\mathbb{R})$ is an isotropic
pseudodifferential operator of order r if and only if its "matrix"
$(K(A))_{m,n} := < A\phi_n,\phi_m>_{L^2(\mathbb{R})}$ is rapidly decreasing
away from the diagonal $\{m = n\}$, order $\frac {r}{2}$ in $m + n$, and where
applying the discrete difference operator along the diagonal decreases the
order by one. Additionally, we use this result to prove an analogue of Beal's
theorem for isotropic pseudodifferential operators.
