Square root of differential operator If $D_x$ is the differential operator. eg. $D_x x^3=3 x^2$.
How can I find out what the operator $Q_x=(1+(k D_x)^2)^{(-1/2)}$ does to a (differentiable) function $f(x)$? ($k$ is a real number)
For instance what is $Q_x x^3$?  

$1+kD^2$ has positive spectrum but this is not enough for the existence of a square root operator on the same space of functions.  This is because derivative is an unbounded operator (so that the expansion of $Q_x$ as a power series in $D$ may not converge), and the square root function is multi-valued.  
The restriction to functions that are polynomials does give a well-defined $Q$ using the power series.
