# Does an isometric automorphism of $L_p (X,\mu, E)$ preserve pointwise convergence?

Let $$(X, \mathcal A, \mu)$$ be a $$\sigma$$-finite measure space and $$(E, |\cdot|)$$ a Banach space. Here we use the Bochner integral. Let $$p \in [1, \infty)$$ and $$q \in (1, \infty]$$ such that $$p^{-1}+q^{-1}=1$$. In an attempt to formalize the ideas in this comment, I have come across below questions, i.e.,

1. Let $$\psi$$ be an isometric automorphism of $$L_{p}(X,\mu, E)$$. Let $$f, f_n \in L_{p}(X,\mu, E)$$ such that $$f_n \to f$$ pointwise $$\mu$$-a.e. Does $$\psi(f_n) \to \psi(f)$$ pointwise $$\mu$$-a.e.?

2. Let $$X^{*}$$ have the Radon-Nikodým property with respect to $$\mu$$. Then there is a canonical isometric isomorphism $$\varphi:L_{p}(X,\mu, E)^* \to L_{q} (X,\mu, E^*)$$ such that $$K (f) = \int_\Omega \langle \varphi(K), f \rangle \mathrm d \mu \quad \forall K \in L_{p}(X,\mu, E)^*, \forall f \in L_{p}(X,\mu, E)$$ Let $$H,H_n \in L_{p}(X,\mu, E)^*$$ such that $$H_n \to H$$ pointwise. Does $$\varphi(H_n) \to \varphi(H)$$ pointwise $$\mu$$-a.e.?

I feel that isometric isomorphisms do not preserve pointwise convergence. However, I could not come up with a counter-example. Could you elaborate on my questions?

I posted this question on MSE, but have not received any satisfying answer. So I post it here.

• Isometries on $L_p$ for $p\not= 2$ are characterized by the Banach-Lamperti theorem. See, e.g., section 2 in Gardella's notes math.chalmers.se/~gardella/Docs/LpOpAlgs.pdf Commented Feb 10, 2023 at 16:42

Point 1) does not hold for the Fourier transform on $$L^2({\bf R}, {\bf C})$$, which is an isometry for a well chosen normalization.
Consider the sequence $$f_n(x) = n {\bf 1}_{[-1/n,1/n]}(x)$$ which converges to 0 for all $$x\neq 0$$. The Fourier transform of $$f_n$$ is proportional to $${sin(x/n) \over {x/n}}$$ which converges to $$1$$ as $$n$$ goes to infinity for all $$x \neq 0$$. And the constant function 1 is not the Fourier transform of 0.