Complex Power of a differential operator Let $(X,\|\cdot\|)$ be a Banach space and consider a sequence $B_n \colon X \to X$ of bounded operators. I remember from my course in operator theory that the partial sum
$$
S_N = \sum^N_{n = 1} B_n
$$
converges in the space $\mathscr{L}(X)$ of bounded operators provided $\sum \|B_n\| < \infty$.
Now I need to consider the complex power $D^{-s}$ of a first order differential, self-adjoint operator $D$, with discrete spectrum. This is defined via the holomorphic functional calculus as the Cauchy integral
$$
D^{-s} := \frac{i}{2\pi} \int_C \lambda^{-s}(D - \lambda)^{-1}\,d\lambda
$$ 
where the contour $C$ is infinite and encloses the spectrum of $D$. 
The question is when is this well - defined? 
In analogy with the series example above, would it be correct to test for absolute convergence of the integral? 
More concretely, since $\|\lambda^{-s}(D - \lambda)^{-1}\| = |\lambda|^{-\Re(s)}\|(D - \lambda)^{-1}\|$ and $\|(D - \lambda)^{-1}\| = O(|\lambda|^{-1})$ as $|\lambda| \to \infty$ this would suggest that the Cauchy integral does make sense provided $\Re(s) > 0$.
Many thanks for your help!

This is a particular case of the complex power of an elliptic operator. In your case the regularization procedure can be made explicit, namely $D^{z}(e_{i})=\lambda_{i}^{z}e_{i}$ for the eigenvectors. So you can think about this as 
$$
D^{z}(\sum a_{i}e_{i})=\sum (a_{i} \lambda_{i}^{z})e_{i}
$$
and whether the later converges obviously has a lot to do with the specific Banach space norm you are using. What you can do is talk about Sobolev spaces and how $D^{z}$ acts on Sobolev spaces. This has been treated systematically in Shubin's book (Chapter 3?). So you should take a look. They proved the operator is well defined for $\Re{z}<0$, for example. 
