please on an infinite dimensional Hilbert space how to difine the essential spectrum of an operator which is negative definite ???
Please help me
Thank you.
2026-03-27 10:38:16.1774607896
Question about the essential spectrum of a negative difinite operator
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The definition of the essential spectrum does not ask for any specific property of the operator. It is just the spectrum of the image of the operator in the Calkin algebra.
More explicitly, $\sigma_{\rm ess}(T)$ consists of those $\lambda\in\sigma(T)$ such that there exists a sequence $\{x_n\}\subset H$ with no convergent subsequence and such that $Tx_n-\lambda x_n\to0$.
Equivalently, $\lambda\in\sigma_{\rm ess}(T)$ if $T-\lambda I$ is not a Fredholm operator (this is pretty much the same as the characterization from the first paragraph).