Let $(H, \langle \cdot, \cdot \rangle)$ be a real Hilbert space and $|\cdot|$ its induced norm. Let $A: D(A) \subset H \to H$ be a maximal monotone (unbounded linear) operator. Let $I:H \to H$ be the identity map. For every $\lambda>0$, set $$ J_\lambda=(I+\lambda A)^{-1} \quad \text { and } \quad A_\lambda=\frac{1}{\lambda}\left(I-J_\lambda\right) ; $$ $J_\lambda$ is called the resolvent of $A$, and $A_\lambda$ is the Yosida approximation (or regularization) of $A$. Then we have
- $\left\|J_\lambda\right\|_{\mathcal{L}(H)} \leq 1$ for $\lambda>0$.
- $\lim_{\lambda \to 0^+} J_\lambda v = v$ for all $v \in H$.
- If $\lambda_n, \lambda >0$ such that $\lambda_n \to \lambda$, then $\|J_{\lambda_n} - J_\lambda\|_{\mathcal L(H)} \to 0$.
On the space $\mathcal L(H)$, we have strong operator topology (SOT), norm operator topology (NOT), and weak operator topology (WOT). We have SOT is stronger than WOT and weaker than NOT.
Now let $\lambda_n >0$ such that $\lambda_n \to 0$.
Does $(J_{\lambda_n})_n$ converge in any of those topologies of $\mathcal L(H)$ as $n \to \infty$? If yes, what is its limit?
Thank you so much for your elaboration!
The convergence in SOT is equivalent to the pointwise convergence which has been already proved
The convergence in WOT will be a consequence of the convergence in SOT.
However, the convergence in NOT is not true in general. Take $H=L^2([0,1], \mathbb C)$, $A= -i\frac{\mathrm d}{\mathrm d x}$ and $v_n = e^{i2n\pi x}$ then
$$Av_n = 2n\pi v_n$$ $$J_{\frac1n}v_n = \frac1{1+2\pi} v_n$$
$$\left\|J_{\frac1n} - I\right\|_{\mathcal L(H)}\ge \left\| J_{\frac1n}v_n - v_n\right\|= \frac{2\pi}{1+2\pi}$$