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

Question

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 $\langle T y_n\mid y_n\rangle$?

Answer 1

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 $.