I recently got some difficulty with my homework question. The question is:
Let $T_1,\dots,T_N$ be a finite collection of bounded linear operators on a hilbert space $H$, each of operator norm $\le 1$.
Suppose that $T_kT_j^\ast = T_k^\ast T_j = 0$ whenever $j \neq k$.
Show that $\displaystyle \sum_{i=1}^N T_i$ satisfies $\|T\| \le 1$.
For the condition $T_k^\ast T_j= 0$, $j \neq k$, I can show $T_k$ and $T_j$ have orthogonal ranges: since $T_k^\ast T_j= 0$, $T_k^\ast T_j f= 0$ for any $f \in H$, so it follows $(T_k^\ast T_j f,g)= 0$ for any $g$. But $(T_k^\ast T_j f,g)=(T_jf,T_kg)=0$, so $T_j$ and $T_k$ should have orthogonal ranges.
However, I cant do anything for condition $T_kT_j^\ast= 0$. But the hint says for $T_kT_j^\ast = 0$, $j \neq k$,introduce the orthogonal projection $P_i$ onto the closure of the range $T_i^\ast$, and show that $T_i f$ = $T_i P_i f$.
I dont quite understand the hint and I need some help with this question.
Beside, the question has a part 1, which is to show if $P_1$ and $P_2$ are two orthogonal projections, with orthogonal ranges, then $P_1 + P_2$ is also an orthogonal projection. I've done this part 1, but I guess this conclusion is helpful for solving this latter part.