Let $H$ be an infinite dimensional seperable Hilbert space. Is there an Irreducible involutive sub algebra $D$ of $B(H)$ with the following properties?:


1)For every open set $U\subset H$ and every Frechet differential map
$f:U \to H$ with $Df(x)\in D,\; \forall x\in U$, the mapping $f$ is automatically $C^{\infty}$


2)For every open set $U\subset H$, a uniform limit $f$ of a sequence of Frechet differential maps $f\_n:U \to H$ with $Df\_n(x)\in D$ is Frechet differentiable which satisfies $Df(x)\in D,\;\forall x\in U$.


The motivation comes from the concepet of holomorphic functions when we put (the real analogy) $ H=\mathbb{R}^2$ and $D=\left \{\begin{pmatrix} a&b\\ -b&a\end{pmatrix} \mid a,b\in \mathbb{R} \right \}$



A kind of holomorphicity of maps on Hilbert space
1. In [this book](https://www.mat.univie.ac.at/%7Emichor/apbookh-ams.pdf), Thm 7.19 (1) $\Longleftrightarrow$ (9) answers this for $D$ the complex linear bounded maps, but you have to assume that $f$ is Gateaux differentiable with the derivative locally Lipschitz.
2. From Thm 7.17 (7) you can deduce an answer, but you have to adapt the definitions and strengthen the convergence at least at one point.


There is also the book:


* Dineen, Seán
Complex analysis in locally convex spaces. (English) Zbl 0484.46044
North-Holland Mathematics Studies, 57. Notas de Matematica (83). Amsterdam - New York - Oxford: North-Holland Publishing Company. XIII, 492 p.


