Let $E$ be a complex Hilbert space, with inner product $\langle\cdot\;, \;\cdot\rangle$ and the norm $\|\cdot\|$. Let $T\in \mathcal{L}(E)$ be bounded linear operators from $E$ to $E$ and $M\in \mathcal{L}(E)^+$ (i.e. $M^*=M$ and $\langle Mx\;, \;x\rangle\geq 0$ for all $x\in E$.

Assume that there exists a sequence $(y_n)\subset E$ such that $\langle My_n\;, \;y_n\rangle=1$.

Does the sequence $(\langle MTy_n\;, \;y_n\rangle)_n$ converge or have a subsequence that converges?

In the case when $M=I$, clearly $(\langle MTy_n\;, \;y_n\rangle)_n$ is bounded, so it has a subsequence that converges.