How can it be proven that the essential numerical range of an operator T is non empty?
Is the essential numerical range of an operator T non empty?
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$\begingroup$
analysis
functional-analysis
2 Answers
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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 $W_e(T)$.
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The essential numerical range contains the essential spectrum and the later is never empty.
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0Does this assume something about the linear operator $T$? Note the assumptions stated in the other Answer. – 2014-03-13