dim $ker(T-\lambda)\le1$ for all $\lambda$ in $\mathbb{C}$ $\iff$ $\exists h\in \mathbb{H} \ s.t. \ h$ is a cyclic vector for $T$.

Question

Let $T$ be a compact normal operator on a $\mathbb{C}$-Hilbert space,$\mathbb{H}$.

dim $ker(T-\lambda)\le1$ for all $\lambda$ in $\mathbb{C}$ $\iff$ $\exists h\in \mathbb{H} \ s.t. \ h$ is a cyclic vector for $T$.

I have figured out the $\Leftarrow$ part.

Let $P$ be the projection onto the closed span of $\{(T-\lambda)^nh;n\ge1\}$, $x=h-Ph$.

Then $h\in \mathbb{C}x\oplus P\mathbb{H}$, $ker(T-\lambda)=r(T-\lambda)^\perp\subseteq P\mathbb{H}^\perp=\mathbb{C}x $

for the $\Rightarrow$ part, I just guess that maybe every $h\ s.t.$ whose projection on every eigenspace is nonzero is a cyclic vector. And I prove that if a vector is a finite linear combination of eigenvectors, then it is not vertical to the closed span of $\{T^nh;n\ge0\}$. So if the range of $T$ is infinite dimensional, what should I do?

Answer 1

By the assumptions, there are distinct non-zero eigenvalues $(\lambda_k)$ with unit eigenvectors $(e_k)$. Let me assume in this answer, that the sequence $(|\lambda_k|)$ is decreasing. This implies $|\lambda_i + \lambda_j| < 2|\lambda_i|$ for all $i\le j$. If $N(T)\ne\{0\}$ then let $e_\infty$ be a unit vector in $N(T)$ otherwise let $e_\infty$ be zero.

Define $$ h:=e_\infty + \sum_{k=1}^\infty \frac1{k^2} e_k. $$ Set $W:=closure(span\{T^jh: \ j\in\mathbb N\})$.

Let me show that $(2\lambda_1)^{-j} (T+\lambda_1I)^jh\rightharpoonup e_1$. For fixed $k$, we have $$ \langle (2\lambda_1)^{-j} (T+\lambda_1I)^jh, e_k\rangle = \frac1{k^2} \left(\frac{\lambda_k+\lambda_1}{2\lambda_1}\right)^j \to \begin{cases} 1 & \text{ if } k=1\\ 0&\text{ if } k>1.\end{cases} $$ In addition $\langle(2\lambda_1)^{-j} (T+\lambda_1I)^jh, e_\infty\rangle=0$ for $j\ge1$. This shows $(2\lambda_1)^{-j} (T+\lambda_1I)^j h\rightharpoonup e_1$, and since $W$ is weakly closed, $e_1\in W$. Similarly, one can show $$ (2\lambda_2)^{-j} ( (T+\lambda_2 I)^j h - (\lambda_1+\lambda_2)^je_1)\rightharpoonup e_2. $$ Then $e_k \in W$ for all $k$, which proves $R(T)\subset W$.

Then it follows $e_\infty = h - \sum_{k=1}^\infty \frac1{k^2} e_k \in W$. This proves $h$ is a cyclic vector.