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 $.

  • $\begingroup$ "Non-obvious", perhaps, since it took fifty years to find a short elementary proof. $\endgroup$ Oct 23 '18 at 1:07
  • $\begingroup$ And now much shorter than Gustafson's are known. Even then, I would still use the word "non-trivial" for the result. $\endgroup$ Oct 23 '18 at 1:15
  • $\begingroup$ @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. $\endgroup$
    – Schüler
    Oct 23 '18 at 6:59
  • $\begingroup$ I don't think it can be written like that. $\endgroup$ Oct 23 '18 at 12:25
  • $\begingroup$ @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. $\endgroup$
    – Schüler
    Oct 24 '18 at 5:44

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