Strong convergence of Spectral Projection

Question

Let $H$ be a Hilbert space and $B(H)$ be the space of all bounded linear operators on $H$.

Assume that $\{A_n\in B(H)\}_n$ strongly converges to $A$.

$E^{|A|}(1,\infty)$ is a spectral projection of $|A|$.

My question is: does $E^{|A_n|}(1,\infty) \rightarrow _{so} E^{|A|}(1,\infty)$ or not?

Answer 1

Let $A_n=\left(1+\frac1n\right)\,I$. Then $A_n\to I$ in norm (and so, also strongly). We have $$ E^{|A_n|}(1,\infty)=I,\ \ \ \ \ \ \ \ E^{|A|}(1,\infty)=0. $$