The following is problem 8 from chapter 4 ("Hilbert Spaces: An Introduction") of Stein and Shakarchi's Real Analysis.
Suppose $\{t_k\}$ is a collection of bounded operators on a Hilbert space $H$. Assume that $$ \|T_kT^*_j\|\leq a_{k-j} $$ and $$ \|T^*_kT_j\|\leq a^*_{k-j} $$ for positive constants $\{a_n\}$ with the property that $\sum_{-\infty}^{\infty} a_n=A<\infty$. Then $S_N(f)$ converges as $N\rightarrow\infty$, for every $f\in H$, with $S_N=\sum_{-N}^{N}T_k$. Moreover, $T=\lim_{N\rightarrow\infty}S_N$ satisfies $\|T\|\leq A$
1.What does $a^*_{k-j}$ mean?
2.Could you please give me a hint how to solve this problem?
Maximiliano: Thank you for your answer, but could you please give me a more detailed explanation. I have found that this question is a generalization of exercise 23 in the same chapter, Perhaps it would be easier to solve or to help me with that problem.