When $H_1$ is the invertible system of the $H_2$ and are connected in parallel manner then what can be said about causality and memory of system.
The output of the system will be $$y = u*(H_1+H_2)$$ Now i think the causality of $H_1+H_2$ can not be determined by previous knowledge of $H_1$ or $H_2$ causality. Please help me to clear my concepts.
Given that $$y = u \star (h_1 + h_2) = u_1 + u_2$$ then, if the individual filters $H_1,H_2$ are causal, then also $H$ is.
This can also be seen from the consideration of the impulsive responses, because $h(n) = h_1(n)+h_2(n)$, then if $h_i(n)=0$ for $n<0$ ($i=1,2$) we conclude $h(n)=0$ for $n<0$.
We cannot conclude the reverse. Given that $H$ is causal, we cannot affirm anything about $H_1,H_2$. Put in other way, it may happen that $H_1$ and $H_2$ are non causal, but "the noncausalities cancel" so that $h(n) = h_1(n)+h_2(n)$ results causal.