bounded operators and rank equal 1

Question

I have a new question: If $X$ is an Hilbert space, and

1. $C\in L(X,\mathbb{C})$
2. $B\in L(\mathbb{C},X)$
3. $(sI-A)^{-1}\in L(X)$ , where $A$ is a generator of an holomorphic semigroup

Can I conclude that $$ rank(C(sI-A)^{-1}B)=1 ? $$

Answer 1

Because $B$ is at most rank-one, and composition cannot increase rank, you have that $$ \operatorname{rank}(C(sI-A)^{-1}B)\leq 1. $$ But a priori you cannot discard that $C(sI-A)^{-1}B$ is zero. Even if $B\ne0$, the range of $(sI-A)^{-1}B$ is one-dimensional; it is enough to choose $C$ such that it is zero on the (one-dimensional) range of $(sI-A)^{-1}B$ to get $C(sI-A)^{-1}B=0$