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The gap of a bounded linear operator on a Hilbert space is defined as $$\operatorname{gap}(A) := \|A\| - r(A),$$ where $r(A)$ denotes the spectral radius of $A$. A natural question to ask is - for which operators is $\operatorname{gap}(A) > 0$ (vs. $\operatorname{gap}(A) = 0$)? A simple but not satisfying fact is that $\operatorname{gap}(A) = 0$ for all normal operators $A$. Even after a detailed search, I could find just one paper that studies gaps to some extent (here.)

Thanks!

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    $\begingroup$ I think you need to consider related operators $B = S^{-1}AS$, where $S$ is invertible. The spectral radius for $B$ is the same as for $A$, but the norm may be different. I believe (I read somewhere long ago?) the infimum of all these norms is the spectral radius. $\endgroup$
    – Gerald Edgar
    Commented Apr 4 at 14:48
  • $\begingroup$ Notice that a nilpotent operator $A$ has $r(A) = 0,$ whereas $\|A\|$ can be arbitrarily large, so I wonder what sort of answer you would regard as useful in your context? $\endgroup$
    – Geoff Robinson
    Commented Apr 5 at 9:18

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