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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 $D=\left \{\begin{pmatrix} a&b\\ -b&a\end{pmatrix} \mid a,b\in \mathbb{R} \right \}$

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  • $\begingroup$ A simple solution would be $D=0$. $\endgroup$
    – user473423
    Apr 7, 2022 at 12:50

1 Answer 1

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  1. In this book, 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.
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  • $\begingroup$ Thank you very much for your answer $\endgroup$ Apr 17, 2022 at 10:26

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