Holomorphic Banach-valued functions and composition I am working on an article about holomorphic functions over complex Banach spaces. To complete a corollary's proof, I was wondering if there exists a result in the following direction:
Let $X,Y$ be complex Banach spaces, $T$ a bounded linear operator from $X$ to $Y$ and $\Gamma: \mathbb{D} \to X$ a holomorphic mapping. Is there some way to asssure that there is a holomorphic function $f: \mathbb{D} \to Y$ with $f(0)=0$, such that $f' = T \circ \Gamma$?
The only thing that I could prove is that $T \circ \Gamma$ is a holomorphic function. I know that a mathematician called Paul Garrett worked on the line of holomorphic vector-valued functions, but I cannot found something specifically related with this.
 A: The map $T$ here is really not relevant; you might as well just replace $\Gamma$ with $T\circ\Gamma$ at the start.  That is, suppose $\Gamma:\mathbb{D}\to Y$ is holomorphic.  Then you want to find a holomorphic $f:\mathbb{D}\to Y$ such that $f(0)=0$ and $f'=\Gamma$.  You can do this in the same way as in the case $Y=\mathbb{C}$: just integrate.  That is, define $$f(z)=\int_0^z\Gamma(s)\,ds$$ where the integral is a Bochner integral of $Y$-valued functions along some path from $0$ to $z$.   To prove that $f'=\Gamma$, you can first show that the integral above is independent of the path chosen by composing with an arbitrary bounded linear functional $\alpha:Y\to\mathbb{C}$ to reduce to the case $Y=\mathbb{C}$ (as discussed in this answer).  Once you have that, you can prove that $f'=\Gamma$ by exactly the same argument as in the case $Y=\mathbb{C}$.
A: @EricWofsey's answer certainly resolves the question at hand.
For context and references: yes, chapters 13, 14, 15 of my fairly recent book (purportedly about automorphic forms, but with lots of background) https://www-users.cse.umn.edu/~garrett/m/v/current_version.pdf
treat such stuff.
The original references would be L. Schwartz' body of work on distribution theory, and A. Grothendieck's early work, on functional analysis. As Bill Casselman helpfully pointed out to me a few years ago, Grothendieck's work about holomorphic vector-valued functions appears in
A. Grothendieck, Sur certaines espaces de
fonctions holomorphes, I, J. Reine Angew. Math. 192 (1953),
35-64; II, 192 (1953), 77-95.
