Is the function mapping a self-adjoint operator on a Hilbert space to the maximum of its spectrum differentiable?

by 0xbadf00d   Last Updated October 10, 2019 05:20 AM

Let $H$ be a $\mathbb R$-Hilbert space. Is $$\left\{A\in\mathfrak L(H):A\text{ is self-adjoint}\right\}\to\mathbb R\;,\;\;\;A\mapsto\max\sigma(A)\tag1$$ differentiable? We know that this is true in the finite-dimensional case $H=\mathbb R^d$, $d\in\mathbb N$. If the claim as stated is not true in general, is there at least a similar result?

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