Sequence of linear operators whose convergence to its pointwise limit is difficult to prove?

Question

I'm asking this purely out of curiosity. Any sequence of linear operators I have seen thus far has a very obvious limit and proving the convergence has not been much of a challenge either. So I'm looking for sequences of linear operators in a normed linear space, whose convergence to their respective pointwise limits is not easy to show. Please share with me any such sequence you have encountered.

Answer 1

Let $H$ be a real or complex Hilbert space and $T \in B(H)$ with $\lVert T \rVert \leqslant 1$, and let's define $A_n$ as $$A_nx:=\frac{1}{n+1}\sum_{k=0}^n T^k x \quad(\forall x \in H)$$ What's the pointwise limit of $A_n$?

(This is one of the extra exercises from the functional analysis course I've had, but the pointwise limit was given for us)

Answer 2

This survey of Michael Lacey discusses the theorem of Carleson, a very difficult one in harmonic analysis, concerning the pointwise convergence of Fourier series:

https://arxiv.org/abs/math/0307008

This question&answers here on Math.SE is a kind of "baby Carleson theorem".