# Operator norm of the sum of a finite collection of bounded linear operator

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.

• Proof of the hint: we have to show that $T_j(I-P_j)f=0$ for each $j$ and $f\in H$. Note that $(I-P_j)f=:g$ is in the orthogonal complement of the range of $T_j^*$. Hence $\langle g,T_j^*h\rangle =0$ for each $h$, and $\langle T_jg,h\rangle=0$. Taking $h=T_jg$, we have what we wanted.
By the first part, $P_1+\dots+P_N$ is a projection, and we can conclude $\lVert Tx\rVert \le \lVert x \rVert \implies \lVert T \rVert \le 1$ since $\lVert (P_1 + \dots P_N)x \rVert \le \lVert x\rVert$.