Let $X$ be a measure space. By Riesz-Fischer, $L^2(X)$ is a Banach space. When $U \subseteq \mathbb C$ is open, we have a notion of holomorphic functions $U \to L^2(X)$. (See the bottom for definitions.) Let's write such functions as $E(s, w)$.
When $X$ has a nice topology (say it a Riemannian manifold) and $E$ is continuous in $(s, w)$ it makes sense to evaluate in $w$ and ask whether the complex-valued $E(s, w_0)$ is holomorphic for a specific $w_0$.
Question. Are there any implications between the holomorphy of $E : U \to L^2(X)$ and the holomorphy of $E(\cdot, w_0) : U \to \mathbb C$ for all $w_0$?
Here's one result in this direction I found: suppose $E : U \to L^2(X)$ is compactly supported, continuous in $(s, w)$, holomorphic in $s$ for all $w$ and $E'$ is continuous in $(s,w)$. Then $E$ and $E'$ are in $L^2$ and we have $$E(s, w) - E(s_0, w) = E'(s_0, w) (s-s_0) + o_w(s-s_0) $$ By assumption the little-$o$ term divided by $s-s_0$ is continuous in $(s,w)$, hence is square integrable and we have $$E(s, w) - E(s_0, w) = E'(s_0, w) (s-s_0) + o(s-s_0) $$ as an $L^2$-statement. I.e. $E : U \to L^2(X)$ is holomorphic.
Equivalent definitions of holomorphic functions $f : U \to Y$, for $Y$ a Banach space are:
- $f$ is everywhere approximately linear: $f(s)-f(s_0) = f'(s_0)(s-s_0) + o(s-s_0)$
- $\lambda \circ f$ is holomorphic for all $\lambda \in Y^*$
- $f$ is locally a power series of the form $\sum A_n (s-s_0)^n$
Note the similarity between 2. and the question, where in a sense we look at $\lambda$ being the evaluation in a point (which is not well-defined on $L^2$).