Proof explanation: Why we can select unit vectors $z_n\in M_n$?

Let $$\mathcal{B}(F)$$ the algebra of all bounded linear operators on a complex Hilbert space $$(F,\langle\cdot\mid\cdot\rangle)$$.

• Let $$T\in \mathcal{B}(F)$$ and assume that there exists sequences of unit vectors $$(x_n)_n$$ and $$(y_n)_n$$ in $$F$$ such that $$\lim_{n\to \infty}\langle T x_n\mid x_n\rangle=\lambda,\;\lim_{n\to \infty}\langle T y_n\mid y_n\rangle= \mu,$$ with $$\lambda$$ and $$\mu$$ are two distinct complex numbers.

• Let $$M_n$$ be a subspace spanned by $$x_n$$ and $$y_n$$ and $$P_n$$ be a projection of $$F$$ onto $${M_n}$$. Consider $$T_n=P_nTP_n$$.

Why we can select unit vectors $$z_n\in M_n$$ such that $$\langle T z_n\mid z_n\rangle$$ is a convex combination of $$\langle T x_n\mid x_n\rangle$$ and $$\langle T y_n\mid y_n\rangle$$?

It's a non-trivial result called the Toeplitz-Hausdorff Theorem. The set $$W(T)=\{\langle Tx,x\rangle:\ \|x\|=1\}$$ is usually called the numerical range of $$T$$. The Toeplitz-Hausdorff Theorem, proven a hundred years ago (exactly, it was in 1918), states that $$W(T)$$ is convex.
If $$P_n$$ is the orthogonal projection (and not just "a" projection) onto $$M_n$$, then you apply Toeplitz-Hausdorff to the operator $$P_nTP_n$$ acting on $$M_n$$ and use that $$\langle T_nx_n,x_n\rangle=\langle P_nTP_nx_n,x_n\rangle$$.
• @MartinArgerami Thank you for your answer. It can be seen that $\langle T x_n\mid x_n\rangle,\langle T y_n\mid y_n\rangle\in W(T_n)$ which is convex so we can select unit vectors $z_n\in F$ such that $\langle T z_n\mid z_n\rangle$ is a convex combination of $\langle T x_n\mid x_n\rangle$ and $\langle T y_n\mid y_n\rangle$. My question is: why $W(T_n)$ can be written as follows: $$W(T_n):=\{\langle T_n x\mid x\rangle;\;x \in M_n\;\;\text{with}\;\|x\|=1\}?$$ Thanks a lot. Oct 23 '18 at 6:59
• @MartinArgerami Since $T_n$ acts on $M_n$, so it should be written as $$W(T_n):=\{\langle T_n x\mid x\rangle;\;x \in M_n\;\;\text{with}\;\|x\|=1\}.$$ Where is my wrong? Thank you for your explanation. Oct 24 '18 at 5:44