0
$\begingroup$

How can it be proven that the essential numerical range of an operator T is non empty?

2 Answers 2

4

I assume $T$ is a bounded operator on a separable infinite-dimensional Hilbert space $\mathcal H$.

$\lambda\in \mathbb C$ belongs to the essential numerical range $W_e(T)$ of $T$ iff there exists an orthonormal sequence $\{e_n\}$ such that $ \langle Te_n, e_n \rangle \to \lambda $

Let $\{e_n\}$ be an orthonormal base of $\mathcal H$. The sequence $ \langle Te_n, e_n \rangle $ is bounded by $\lVert T \rVert$ and so you can extract a convergent subsequence from it. Its limit belongs to $W_e(T)$.

0

The essential numerical range contains the essential spectrum and the later is never empty.

  • 0
    Does this assume something about the linear operator $T$? Note the assumptions stated in the other Answer.2014-03-13