# equivalent definition of k-rank numerical range of an operator

Let $$T\in\mathscr{B(\mathcal{H})}$$ where $$\mathcal{H}$$ is an infinite dimensional seperable Hilbert Space and $$k\in\mathbb{N}\cup\{\infty\}$$. Now we define k-rank numerical range of $$T$$ denoted by $$\Lambda_k(T)$$ is defined as $$\Lambda_k(T):=\{\lambda\in\mathbb{C}: PTP=\lambda P, \text{ for some orthogonal projection } P \text{ of rank } k\}$$ or equivalently we can write $$\lambda\in\Lambda_k(T) \text{ iff there exists an orthonormal set } \{f_j\}_{j=1}^k \text{ s.t. } \langle Tf_j,f_r\rangle=\lambda\delta_{j,r} \text{ for } j,r\in\{1,2\ldots, k\}$$ where $$\delta_{j,r}$$ is Kronecker delta. Clearly $$\Lambda_1(T)=W(T)$$ i.e. $$\Lambda_k(T)$$ is a genaralization of numerical range $$W(T)$$.

Question: Let $$T\in\mathscr{B(\mathcal{H})}$$ be a normal operator. Show that if $$\Re(e^{i\theta}\lambda)\in\Lambda_k(\Re(e^{i\theta}T))\text{ for all }\theta\in[0,2\pi)\text{ then }\lambda\in\Lambda_k(T)$$

Comments: I could able to prove this using some big result of k-rank numerical range of an operator in terms of intersection of half-planes. But unfortunately I could not able to prove it using just definition of k-rank numerical range (written in the beginning) which I feel one could. This result is easy to see for numerical range $$W(T)=\Lambda_1(T)$$.

Any Hints or comments is highly appreciated. Instead of showing $$\lambda\in\Lambda_k(T)$$, if one show $$\lambda\in\overline{\Lambda_k(T)}$$ is also highly appreciable.