Reference request: Laurent series of vector-valued function

Question

I would like a reference to the proof of the following theorem stated in "An Invitation to Operator Theory by Abramovich and Aliprantis":

Let $X$ be a Banach space and $O \subseteq\mathbb{C}$ be an open set. Further, let $f: O \to X$ be an analytic function and $A_{R_1,R_2}(\lambda_0)\subseteq O$ be an open annulus.Then $f$ has a unique Laurent series expansion on $A_{R_1,R_2}(\lambda_0).$

I have actually been using Laurent series expansion of resolvent of a closed operator and have been wondering to see it actually exists.

Answer 1

Try the book Holomorphic Operator Functions of One Variable and Applications, Israel Gohberg/Jurgen Leiterer, starting from page 23.

Answer 2

Suppose $f : A(R_1,R_2) \rightarrow X$ is holomorphic. For $R_1 < r_1 < r_2 < R_2$, the Cauchy integral for $f$ gives $$ f(z) = \frac{1}{2\pi i}\oint_{|w|=r_2}\frac{1}{w-z}f(w)dw-\frac{1}{2\pi i}\oint_{|w|=r_1}\frac{1}{w-z}f(w)dw. $$ Then the standard technique used to expand the first integral in a series of non-negative powers of $z$ and to expand the second integral in a series of negative powers of $z$ works for this vector function as well. So you obtain the desired vector Laurent series that converges in $r_1 < |z| < r_2$ for all $R_1 < r_1 < r_2 < R_2$. Hence the Laurent series converges in $R_1 < |z| < R_2$.