I am in process of writing up a document and would like to refer to a couple basic facts on vector-valued holomorphic functions. In particular, I am looking for a reference (book or expository article) where versions of below statements are proved.
A:
Let $\Omega$ be a domain in $\mathbb{C}$, $X$ be a Banach space, and $B(X)$ the $C^*$-algebra of bounded linear operators on $X$. A function $F: \Omega \to B(X)$ is holomorphic in the vector-valued sense, i.e, for every $z \in \Omega$ we have that the limit $$\lim_{h \to 0} \frac{F(z+h)-F(z)}{h}$$ exists in the norm-topology on $B(X)$, if and only if the function $$\Omega \ni z \to \langle \phi, F(z)x \rangle \in \mathbb{C}$$ is a scalar-valued holomorphic function, for every continuous linear functional $\phi$ on $X$ and every $x \in X$, where $\langle \cdot, \cdot \rangle$ is the usual duality pairing between $X^*$ and $X$.
B:
Let $\sum_{n=0}^\infty A_n z^n$ be a formal power series, where $A_n: X \to X$ are bounded linear operators on a Banach space $X$. If the power series converges on some (open) subset $D \subset \mathbb{C}$ in the weak/strong operator topology sense, then it converges in the norm topology.
A good reference is:
A.E. Taylor and D.C. Lay: Introduction to functional analysis (Wiley &Sons)
(Chapter V)