Let $A_n$ and $A$ be bounded self-adjoint operators in a Hilbert space, such that $A_n\to A$ strongly. Then it is well known that $(z-A_n)^{-1}\to(z-A)^{-1}$ strongly for each $z\in\mathbb{C}\setminus\mathbb{R}$, and even strong convergence of $f(A_n)$ to $f(A)$ for nice enough functions $f$.

My question is a bit similar to this, but involves series, instead of sequences. Let $T_n$ and $T$ be bounded self-adjoint operators, such that $$ T_1+\ldots+T_n\to T \qquad \textrm{strongly}. $$ Is it true that $$ \frac{T_1}{z-T_1} + \ldots + \frac{T_n}{z-T_n} , $$ converges strongly to some self-adjoint operator? Do we have convergence of $$ f(T_1)T_1+\ldots+f(T_n)T_n , $$ for nice enough functions? A particular case of interest is $f$ being the indicator function of the positive half line. If it helps, $T$ can be assumed to be invertible.

Regarding the aforementioned expectations:

- Are they hopelessly false?
- If it is salvageable, what kind of extra conditions does one need?
- What is the correct keyword to look up?