# Controlling norm of operators sending a fixed vector to another

Let $$A$$ be a $$C^*$$-algebra acting on a Hilbert space that admits a cyclic unit vector $$\xi \in H$$. Pose $$S_\xi = \{\eta \in A \xi: \| \eta \| = 1\}$$, and for each $$\eta \in S_\xi$$, pose $$A_\eta = \{a \in A : a \xi = \eta\}$$, so $$A_\eta \ne \emptyset$$ by definition; and let $$c_\eta = \inf\{\| a \| : a \in A_\eta\}$$. Clearly, $$c_\eta \ge 1$$ for all $$\eta \in S_\xi$$. I am interested in the following questions.

Question 1. Is it true that $$c_\eta = 1$$ for all $$\eta \in S_\xi$$ ?

Question 1 (weak form). Is it true that $$S(A)\xi$$ is dense in $$S(H)$$, where $$S(X)$$ denotes the closed unit ball of a normed space $$X$$?

Question 2. If the answer of Question 1 turns out to be negative, can there be any examples for which $$\sup\{c_\eta : \eta \in S_\xi\} = +\infty$$ ?

Question 3. What if we drop the assumption of $$\xi$$ being cyclic but merely asks $$A$$ acts non-degenerately on $$H$$ ?

Set $$A = C([0,1])$$ and $$H = L^2([0,1])$$ (with respect to Lebesgue measure) with $$\xi=1$$ the constant function. Then $$A\xi$$ is the image of $$C([0,1])$$ in $$L^2([0,1])$$, which is a norm-decreasing injective but not bounded below inclusion. Then $$S_\xi$$ is the intersection of the unit ball of $$L^2$$ with the continuous functions, and for each $$\eta$$ we see that $$A_\eta$$ is the singleton $$\{\eta\}$$.
Thus, Q1 (both versions) have a negative answer, this example gives a positive answer to Q2, and Q3 can be reduced to the main question by restricting the action of $$A$$ to the cyclic subspace generated by $$\xi$$.
However, if $$A$$ were the compact operators on $$H$$ then we'd have a positive answer to Q1.
Consider $$H=L^2([0,1])$$ and $$A = C([0,1])$$ acting on $$H$$ by multiplication. Then $$\xi = 1$$, the constant function $$1$$, is a cyclic vector. Clearly $$A\xi = C([0,1])$$, considered as a subset of $$L^2([0,1])$$ so $$S_{\xi}$$ is just the set of all continuous functions on $$[0,1]$$ with unit $$L^2$$-norm. For each $$\eta\in S_{\xi}$$, the set $$A_{\eta}$$ is a singleton $$\{\eta\}$$, considered as the multiplication operator by $$\eta$$ on $$L^2([0,1])$$. Therefore, $$c_{\eta} = \|\eta\|_{\infty}$$. Recall that on the other hand, $$\|\eta\|_{2} = 1$$. From here, Question 2 has an affirmative answer because you can make the sup norm as large as you want while keeping $$L^2$$-norm unchanged.