Holomorphic functional Calculus in Dunford and Schwartz I am currently studying the spectral theory for bounded operators as described in the book "Linear Operators" by Dunford and Schwartz because I would like to obtain a better understanding of the functional Calculus that uses the Cauchy formula
$$
f(T) = \frac{1}{2 \pi i} \int_C f(\lambda)(\lambda - T)^{-1} \,d\lambda
$$
where $T$ is a bounded operator, $f$ is a function of a complex variable $\lambda$ which is analytic in an open subset of $\mathbb{C}$ that contains the spectrum $\sigma(T)$ of $T$, and $C$ is an appropriately chosen contour.
Now the text starts with the case of operators on finite dimensional spaces, and it is there where I got stuck with the following statement:
For $f$ a function as described above, let $P$ be a polynomial such that 
$$
f^{(m)}(\lambda) = P^{(m)}(\lambda) \,, \qquad m \le \nu(\lambda) - 1 \,,
$$
for each $\lambda \in \sigma(T)$. 
(The number $\nu(\lambda)$, where $\lambda \in \sigma(T)$, denotes the least integer for such that $(\lambda I - T)^\nu x = 0$ for every vector $x$ that satisfies $(\lambda I - T)^{\nu + 1}x = 0$.)
How can I find such a polynomial? I thought I should use the power series expansion of $f$, but how do I ensure that the equation above holds for each $\lambda \in \sigma(T)$?
Thanks a lot for your help!!

It looks like what you need is Hermite Interpolation. It requires you to prescribe the same number of derivatives at all points; but you are dealing with a finite number of points, so you just take the bigger $m$ and make up the values for the missing derivatives. 
