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Let $H$ be a complex Hilbert space and let $T \in \mathcal{B}(H)$ be a linear, bounded operator. Given $x \in H$ we define its local spectral radius as $$r_T(x) = \limsup\limits_{n\rightarrow\infty} \|T^nx\|^{1/n}.$$ Now, let $(x_n)_n$ be a sequence converging to $x$. I've been looking for any results to assure the convergence of the local spectral radius, id est, $r_T(x_n) \rightarrow r_T(x)$. Obviously this convergence cannot happen for every operator and for every vector, but I would like to know if anyone knows any condition to assure this convergence.

Thank you very much.

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    $\begingroup$ I don't know if the terminology is standard but if not, I'd rather call it "pointwise spectral radius". $\endgroup$
    – YCor
    Commented Mar 11, 2020 at 11:02
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    $\begingroup$ @YCor: Actually, the terminology is standard. There's an entire field called "local spectral theory" that deals with local - in the sense of "pointwise" - spectral properties. $\endgroup$
    – Jochen Glueck
    Commented Mar 11, 2020 at 13:54

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