I am trying to figure out convergence conditions for the series $B\equiv \sum_{n=0}^\infty A_n A_n'$ where $A_n = \sum_{k=0}^\infty a_{n+k}$ and $A_n'$ being the transpose of $A_n$.
I have a number of related questions.
1/ In the scalar case, the Cauchy product $A_nA_n$ converges if $\sum_k |a_k|$ is convergent. Does absolute convergence also suffices to guarantees that $B$ converges? If not, what does?
2/ Does this generalize to the case where $a_k$ are matrices?
3/ Finally, I am ultimately interested in convergence of $C\equiv \sum_{n=0}^\infty \lambda^n A_n A_n'$ for some $\lambda>1$ where $\lambda$ can be arbitrary close to 1. Is it true that if $B$ converges, then there exists some $\lambda>1$ so that also $C$ converges?