Let $(E, \langle\cdot\;, \;\cdot\rangle)$ be a complex Hilbert space. Let $T\in\mathcal{L}(E)$ and $M\in \mathcal{L}(E)^+$.

Assume that $T(ker(M))\nsubseteq ker(M)$. We define the following subset:

\begin{eqnarray*} S_M(T) &=&\{\lambda\in \mathbb{C}\,;\;\; \exists\,(\alpha_n,\beta_n)\in ker(M)\times \overline{Im(M)}\,;\;\;\|M^{1/2}\beta_n\|=1, \displaystyle\lim_{n\rightarrow+\infty}\langle MT \alpha_n\; |\;\beta_n\rangle+\langle MT \beta_n\; |\;\beta_n\rangle=\lambda,\\ &&\phantom{+++++}\;\hbox{and}\;\displaystyle\lim_{n\rightarrow+\infty}\|M^{1/2} T(\alpha_n+\beta_n)\|<\infty\;\}. \end{eqnarray*} What do you think about the convexity of $S_M(T)$?? I try with an example of $M$ and $T$ such that $T(ker(M))\nsubseteq ker(M)$, I get $S_M(T)=\mathbb{C}$.

I claim that $S_M(T)=\mathbb{C}$. Do you think that my claim is true?

Thank you for your help!!